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Theorem infxpenlem 9026
Description: Lemma for infxpen 9027. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
leweon.1 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
r0weon.1 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
infxpen.1 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
infxpen.2 (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)))
infxpen.3 𝑀 = ((1st𝑤) ∪ (2nd𝑤))
infxpen.4 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))
Assertion
Ref Expression
infxpenlem ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Distinct variable groups:   𝐴,𝑎   𝑤,𝐽   𝑧,𝑤,𝐿   𝑧,𝑚,𝑀   𝜑,𝑤,𝑧   𝑧,𝑄   𝑚,𝑎,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚,𝑎)   𝐴(𝑥,𝑦,𝑧,𝑤,𝑚)   𝑄(𝑥,𝑦,𝑤,𝑚,𝑎)   𝑅(𝑥,𝑦,𝑧,𝑤,𝑚,𝑎)   𝐽(𝑥,𝑦,𝑧,𝑚,𝑎)   𝐿(𝑥,𝑦,𝑚,𝑎)   𝑀(𝑥,𝑦,𝑤,𝑎)

Proof of Theorem infxpenlem
StepHypRef Expression
1 sseq2 3768 . . . 4 (𝑎 = 𝑚 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝑚))
2 xpeq12 5291 . . . . . 6 ((𝑎 = 𝑚𝑎 = 𝑚) → (𝑎 × 𝑎) = (𝑚 × 𝑚))
32anidms 680 . . . . 5 (𝑎 = 𝑚 → (𝑎 × 𝑎) = (𝑚 × 𝑚))
4 id 22 . . . . 5 (𝑎 = 𝑚𝑎 = 𝑚)
53, 4breq12d 4817 . . . 4 (𝑎 = 𝑚 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝑚 × 𝑚) ≈ 𝑚))
61, 5imbi12d 333 . . 3 (𝑎 = 𝑚 → ((ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎) ↔ (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)))
7 sseq2 3768 . . . 4 (𝑎 = 𝐴 → (ω ⊆ 𝑎 ↔ ω ⊆ 𝐴))
8 xpeq12 5291 . . . . . 6 ((𝑎 = 𝐴𝑎 = 𝐴) → (𝑎 × 𝑎) = (𝐴 × 𝐴))
98anidms 680 . . . . 5 (𝑎 = 𝐴 → (𝑎 × 𝑎) = (𝐴 × 𝐴))
10 id 22 . . . . 5 (𝑎 = 𝐴𝑎 = 𝐴)
119, 10breq12d 4817 . . . 4 (𝑎 = 𝐴 → ((𝑎 × 𝑎) ≈ 𝑎 ↔ (𝐴 × 𝐴) ≈ 𝐴))
127, 11imbi12d 333 . . 3 (𝑎 = 𝐴 → ((ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎) ↔ (ω ⊆ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)))
13 infxpen.2 . . . . . . . 8 (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)))
14 vex 3343 . . . . . . . . . . . . 13 𝑎 ∈ V
1514, 14xpex 7127 . . . . . . . . . . . 12 (𝑎 × 𝑎) ∈ V
16 simpll 807 . . . . . . . . . . . . . . . . . 18 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
1713, 16sylbi 207 . . . . . . . . . . . . . . . . 17 (𝜑𝑎 ∈ On)
18 onss 7155 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ On → 𝑎 ⊆ On)
1917, 18syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑎 ⊆ On)
20 xpss12 5281 . . . . . . . . . . . . . . . 16 ((𝑎 ⊆ On ∧ 𝑎 ⊆ On) → (𝑎 × 𝑎) ⊆ (On × On))
2119, 19, 20syl2anc 696 . . . . . . . . . . . . . . 15 (𝜑 → (𝑎 × 𝑎) ⊆ (On × On))
22 leweon.1 . . . . . . . . . . . . . . . . 17 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}
23 r0weon.1 . . . . . . . . . . . . . . . . 17 𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}
2422, 23r0weon 9025 . . . . . . . . . . . . . . . 16 (𝑅 We (On × On) ∧ 𝑅 Se (On × On))
2524simpli 476 . . . . . . . . . . . . . . 15 𝑅 We (On × On)
26 wess 5253 . . . . . . . . . . . . . . 15 ((𝑎 × 𝑎) ⊆ (On × On) → (𝑅 We (On × On) → 𝑅 We (𝑎 × 𝑎)))
2721, 25, 26mpisyl 21 . . . . . . . . . . . . . 14 (𝜑𝑅 We (𝑎 × 𝑎))
28 weinxp 5343 . . . . . . . . . . . . . 14 (𝑅 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
2927, 28sylib 208 . . . . . . . . . . . . 13 (𝜑 → (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
30 infxpen.1 . . . . . . . . . . . . . 14 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
31 weeq1 5254 . . . . . . . . . . . . . 14 (𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) → (𝑄 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎)))
3230, 31ax-mp 5 . . . . . . . . . . . . 13 (𝑄 We (𝑎 × 𝑎) ↔ (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) We (𝑎 × 𝑎))
3329, 32sylibr 224 . . . . . . . . . . . 12 (𝜑𝑄 We (𝑎 × 𝑎))
34 infxpen.4 . . . . . . . . . . . . 13 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))
3534oiiso 8607 . . . . . . . . . . . 12 (((𝑎 × 𝑎) ∈ V ∧ 𝑄 We (𝑎 × 𝑎)) → 𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
3615, 33, 35sylancr 698 . . . . . . . . . . 11 (𝜑𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
37 isof1o 6736 . . . . . . . . . . 11 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎))
38 f1ocnv 6310 . . . . . . . . . . 11 (𝐽:dom 𝐽1-1-onto→(𝑎 × 𝑎) → 𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽)
39 f1of1 6297 . . . . . . . . . . 11 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽)
4036, 37, 38, 394syl 19 . . . . . . . . . 10 (𝜑𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽)
41 f1f1orn 6309 . . . . . . . . . 10 (𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽𝐽:(𝑎 × 𝑎)–1-1-onto→ran 𝐽)
4215f1oen 8142 . . . . . . . . . 10 (𝐽:(𝑎 × 𝑎)–1-1-onto→ran 𝐽 → (𝑎 × 𝑎) ≈ ran 𝐽)
4340, 41, 423syl 18 . . . . . . . . 9 (𝜑 → (𝑎 × 𝑎) ≈ ran 𝐽)
44 f1ofn 6299 . . . . . . . . . . 11 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽 Fn (𝑎 × 𝑎))
4536, 37, 38, 444syl 19 . . . . . . . . . 10 (𝜑𝐽 Fn (𝑎 × 𝑎))
4636adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)))
4737, 38, 393syl 18 . . . . . . . . . . . . . . . . . 18 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽)
48 cnvimass 5643 . . . . . . . . . . . . . . . . . . 19 (𝑄 “ {𝑤}) ⊆ dom 𝑄
49 inss2 3977 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
5030, 49eqsstri 3776 . . . . . . . . . . . . . . . . . . . . 21 𝑄 ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎))
51 dmss 5478 . . . . . . . . . . . . . . . . . . . . 21 (𝑄 ⊆ ((𝑎 × 𝑎) × (𝑎 × 𝑎)) → dom 𝑄 ⊆ dom ((𝑎 × 𝑎) × (𝑎 × 𝑎)))
5250, 51ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 dom 𝑄 ⊆ dom ((𝑎 × 𝑎) × (𝑎 × 𝑎))
53 dmxpid 5500 . . . . . . . . . . . . . . . . . . . 20 dom ((𝑎 × 𝑎) × (𝑎 × 𝑎)) = (𝑎 × 𝑎)
5452, 53sseqtri 3778 . . . . . . . . . . . . . . . . . . 19 dom 𝑄 ⊆ (𝑎 × 𝑎)
5548, 54sstri 3753 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎)
56 f1ores 6312 . . . . . . . . . . . . . . . . . 18 ((𝐽:(𝑎 × 𝑎)–1-1→dom 𝐽 ∧ (𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎)) → (𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})))
5747, 55, 56sylancl 697 . . . . . . . . . . . . . . . . 17 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → (𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})))
5815, 15xpex 7127 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 × 𝑎) × (𝑎 × 𝑎)) ∈ V
5958inex2 4952 . . . . . . . . . . . . . . . . . . . . 21 (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) ∈ V
6030, 59eqeltri 2835 . . . . . . . . . . . . . . . . . . . 20 𝑄 ∈ V
6160cnvex 7278 . . . . . . . . . . . . . . . . . . 19 𝑄 ∈ V
6261imaex 7269 . . . . . . . . . . . . . . . . . 18 (𝑄 “ {𝑤}) ∈ V
6362f1oen 8142 . . . . . . . . . . . . . . . . 17 ((𝐽 ↾ (𝑄 “ {𝑤})):(𝑄 “ {𝑤})–1-1-onto→(𝐽 “ (𝑄 “ {𝑤})) → (𝑄 “ {𝑤}) ≈ (𝐽 “ (𝑄 “ {𝑤})))
6446, 57, 633syl 18 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≈ (𝐽 “ (𝑄 “ {𝑤})))
65 sseqin2 3960 . . . . . . . . . . . . . . . . . . 19 ((𝑄 “ {𝑤}) ⊆ (𝑎 × 𝑎) ↔ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤}))
6655, 65mpbi 220 . . . . . . . . . . . . . . . . . 18 ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤})) = (𝑄 “ {𝑤})
6766imaeq2i 5622 . . . . . . . . . . . . . . . . 17 (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽 “ (𝑄 “ {𝑤}))
68 isocnv 6743 . . . . . . . . . . . . . . . . . . . 20 (𝐽 Isom E , 𝑄 (dom 𝐽, (𝑎 × 𝑎)) → 𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽))
6946, 68syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽))
70 simpr 479 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑤 ∈ (𝑎 × 𝑎))
71 isoini 6751 . . . . . . . . . . . . . . . . . . 19 ((𝐽 Isom 𝑄, E ((𝑎 × 𝑎), dom 𝐽) ∧ 𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
7269, 70, 71syl2anc 696 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})))
73 fvex 6362 . . . . . . . . . . . . . . . . . . . . 21 (𝐽𝑤) ∈ V
7473epini 5653 . . . . . . . . . . . . . . . . . . . 20 ( E “ {(𝐽𝑤)}) = (𝐽𝑤)
7574ineq2i 3954 . . . . . . . . . . . . . . . . . . 19 (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (dom 𝐽 ∩ (𝐽𝑤))
7634oicl 8599 . . . . . . . . . . . . . . . . . . . . 21 Ord dom 𝐽
77 f1of 6298 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐽:(𝑎 × 𝑎)–1-1-onto→dom 𝐽𝐽:(𝑎 × 𝑎)⟶dom 𝐽)
7836, 37, 38, 774syl 19 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝐽:(𝑎 × 𝑎)⟶dom 𝐽)
7978ffvelrnda 6522 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ dom 𝐽)
80 ordelss 5900 . . . . . . . . . . . . . . . . . . . . 21 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ⊆ dom 𝐽)
8176, 79, 80sylancr 698 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ⊆ dom 𝐽)
82 sseqin2 3960 . . . . . . . . . . . . . . . . . . . 20 ((𝐽𝑤) ⊆ dom 𝐽 ↔ (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8381, 82sylib 208 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ (𝐽𝑤)) = (𝐽𝑤))
8475, 83syl5eq 2806 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (dom 𝐽 ∩ ( E “ {(𝐽𝑤)})) = (𝐽𝑤))
8572, 84eqtrd 2794 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ ((𝑎 × 𝑎) ∩ (𝑄 “ {𝑤}))) = (𝐽𝑤))
8667, 85syl5eqr 2808 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽 “ (𝑄 “ {𝑤})) = (𝐽𝑤))
8764, 86breqtrd 4830 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≈ (𝐽𝑤))
8887ensymd 8172 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≈ (𝑄 “ {𝑤}))
89 infxpen.3 . . . . . . . . . . . . . . . . . . 19 𝑀 = ((1st𝑤) ∪ (2nd𝑤))
90 fvex 6362 . . . . . . . . . . . . . . . . . . . 20 (1st𝑤) ∈ V
91 fvex 6362 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑤) ∈ V
9290, 91unex 7121 . . . . . . . . . . . . . . . . . . 19 ((1st𝑤) ∪ (2nd𝑤)) ∈ V
9389, 92eqeltri 2835 . . . . . . . . . . . . . . . . . 18 𝑀 ∈ V
9493sucex 7176 . . . . . . . . . . . . . . . . 17 suc 𝑀 ∈ V
9594, 94xpex 7127 . . . . . . . . . . . . . . . 16 (suc 𝑀 × suc 𝑀) ∈ V
96 xpss 5282 . . . . . . . . . . . . . . . . . . . 20 (𝑎 × 𝑎) ⊆ (V × V)
97 simp3 1133 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (𝑄 “ {𝑤}))
98 vex 3343 . . . . . . . . . . . . . . . . . . . . . . 23 𝑤 ∈ V
99 vex 3343 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 ∈ V
10099eliniseg 5652 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤 ∈ V → (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤))
10198, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑄 “ {𝑤}) ↔ 𝑧𝑄𝑤)
10297, 101sylib 208 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧𝑄𝑤)
10330breqi 4810 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑄𝑤𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤)
104 brin 4856 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧(𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
105103, 104bitri 264 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑄𝑤 ↔ (𝑧𝑅𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤))
106105simprbi 483 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤)
107 brxp 5304 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤 ↔ (𝑧 ∈ (𝑎 × 𝑎) ∧ 𝑤 ∈ (𝑎 × 𝑎)))
108107simplbi 478 . . . . . . . . . . . . . . . . . . . . 21 (𝑧((𝑎 × 𝑎) × (𝑎 × 𝑎))𝑤𝑧 ∈ (𝑎 × 𝑎))
109102, 106, 1083syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (𝑎 × 𝑎))
11096, 109sseldi 3742 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (V × V))
11117adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ On)
1121113adant3 1127 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑎 ∈ On)
113 xp1st 7365 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (1st𝑧) ∈ 𝑎)
114 onelon 5909 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (1st𝑧) ∈ 𝑎) → (1st𝑧) ∈ On)
115113, 114sylan2 492 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (1st𝑧) ∈ On)
116112, 109, 115syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ On)
117 eloni 5894 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 ∈ On → Ord 𝑎)
118 elxp7 7368 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑤 ∈ (𝑎 × 𝑎) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎)))
119118simprbi 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑤 ∈ (𝑎 × 𝑎) → ((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎))
120 ordunel 7192 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Ord 𝑎 ∧ (1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎)
1211203expib 1117 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Ord 𝑎 → (((1st𝑤) ∈ 𝑎 ∧ (2nd𝑤) ∈ 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎))
122117, 119, 121syl2im 40 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 ∈ On → (𝑤 ∈ (𝑎 × 𝑎) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎))
123111, 70, 122sylc 65 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((1st𝑤) ∪ (2nd𝑤)) ∈ 𝑎)
12489, 123syl5eqel 2843 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑀𝑎)
125 simprr 813 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 𝑚𝑎)
12613, 125sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ∀𝑚𝑎 𝑚𝑎)
127 simprl 811 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
12813, 127sylbi 207 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ω ⊆ 𝑎)
129 iscard 8991 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 ↔ (𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎))
130 cardlim 8988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎))
131 sseq2 3768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (ω ⊆ (card‘𝑎) ↔ ω ⊆ 𝑎))
132 limeq 5896 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((card‘𝑎) = 𝑎 → (Lim (card‘𝑎) ↔ Lim 𝑎))
133131, 132bibi12d 334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((card‘𝑎) = 𝑎 → ((ω ⊆ (card‘𝑎) ↔ Lim (card‘𝑎)) ↔ (ω ⊆ 𝑎 ↔ Lim 𝑎)))
134130, 133mpbii 223 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((card‘𝑎) = 𝑎 → (ω ⊆ 𝑎 ↔ Lim 𝑎))
135129, 134sylbir 225 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → (ω ⊆ 𝑎 ↔ Lim 𝑎))
136135biimpa 502 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) ∧ ω ⊆ 𝑎) → Lim 𝑎)
13717, 126, 128, 136syl21anc 1476 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → Lim 𝑎)
138137adantr 472 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → Lim 𝑎)
139 limsuc 7214 . . . . . . . . . . . . . . . . . . . . . . . 24 (Lim 𝑎 → (𝑀𝑎 ↔ suc 𝑀𝑎))
140138, 139syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑀𝑎 ↔ suc 𝑀𝑎))
141124, 140mpbid 222 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
142 onelon 5909 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ suc 𝑀𝑎) → suc 𝑀 ∈ On)
14317, 141, 142syl2an2r 911 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀 ∈ On)
1441433adant3 1127 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → suc 𝑀 ∈ On)
145 ssun1 3919 . . . . . . . . . . . . . . . . . . . . 21 (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
146145a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
147105simplbi 478 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑄𝑤𝑧𝑅𝑤)
148 df-br 4805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧𝑅𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑅)
14923eleq2i 2831 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑧, 𝑤⟩ ∈ 𝑅 ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))})
150 opabid 5132 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))} ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
151148, 149, 1503bitri 286 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧𝑅𝑤 ↔ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤))))
152151simprbi 483 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧𝑅𝑤 → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))
153 simpl 474 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤) → ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)))
154153orim2i 541 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)) → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
155152, 154syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧𝑅𝑤 → (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
156 fvex 6362 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1st𝑧) ∈ V
157 fvex 6362 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2nd𝑧) ∈ V
158156, 157unex 7121 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑧) ∪ (2nd𝑧)) ∈ V
159158elsuc 5955 . . . . . . . . . . . . . . . . . . . . . . 23 (((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)) ↔ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ ((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤))))
160155, 159sylibr 224 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc ((1st𝑤) ∪ (2nd𝑤)))
161 suceq 5951 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 = ((1st𝑤) ∪ (2nd𝑤)) → suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤)))
16289, 161ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑀 = suc ((1st𝑤) ∪ (2nd𝑤))
163160, 162syl6eleqr 2850 . . . . . . . . . . . . . . . . . . . . 21 (𝑧𝑅𝑤 → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
164102, 147, 1633syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)
165 ontr2 5933 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (1st𝑧) ∈ suc 𝑀))
166165imp 444 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((1st𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (1st𝑧) ∈ suc 𝑀)
167116, 144, 146, 164, 166syl22anc 1478 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (1st𝑧) ∈ suc 𝑀)
168 xp2nd 7366 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ (𝑎 × 𝑎) → (2nd𝑧) ∈ 𝑎)
169 onelon 5909 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ On ∧ (2nd𝑧) ∈ 𝑎) → (2nd𝑧) ∈ On)
170168, 169sylan2 492 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ On ∧ 𝑧 ∈ (𝑎 × 𝑎)) → (2nd𝑧) ∈ On)
171112, 109, 170syl2anc 696 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ On)
172 ssun2 3920 . . . . . . . . . . . . . . . . . . . . 21 (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧))
173172a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)))
174 ontr2 5933 . . . . . . . . . . . . . . . . . . . . 21 (((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) → (((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀) → (2nd𝑧) ∈ suc 𝑀))
175174imp 444 . . . . . . . . . . . . . . . . . . . 20 ((((2nd𝑧) ∈ On ∧ suc 𝑀 ∈ On) ∧ ((2nd𝑧) ⊆ ((1st𝑧) ∪ (2nd𝑧)) ∧ ((1st𝑧) ∪ (2nd𝑧)) ∈ suc 𝑀)) → (2nd𝑧) ∈ suc 𝑀)
176171, 144, 173, 164, 175syl22anc 1478 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → (2nd𝑧) ∈ suc 𝑀)
177 elxp7 7368 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ (suc 𝑀 × suc 𝑀) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)))
178177biimpri 218 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ suc 𝑀 ∧ (2nd𝑧) ∈ suc 𝑀)) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
179110, 167, 176, 178syl12anc 1475 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎) ∧ 𝑧 ∈ (𝑄 “ {𝑤})) → 𝑧 ∈ (suc 𝑀 × suc 𝑀))
1801793expia 1115 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑧 ∈ (𝑄 “ {𝑤}) → 𝑧 ∈ (suc 𝑀 × suc 𝑀)))
181180ssrdv 3750 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ⊆ (suc 𝑀 × suc 𝑀))
182 ssdomg 8167 . . . . . . . . . . . . . . . 16 ((suc 𝑀 × suc 𝑀) ∈ V → ((𝑄 “ {𝑤}) ⊆ (suc 𝑀 × suc 𝑀) → (𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀)))
18395, 181, 182mpsyl 68 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀))
184128adantr 472 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ω ⊆ 𝑎)
185 nnfi 8318 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
186 xpfi 8396 . . . . . . . . . . . . . . . . . . . . . 22 ((suc 𝑀 ∈ Fin ∧ suc 𝑀 ∈ Fin) → (suc 𝑀 × suc 𝑀) ∈ Fin)
187186anidms 680 . . . . . . . . . . . . . . . . . . . . 21 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ∈ Fin)
188 isfinite 8722 . . . . . . . . . . . . . . . . . . . . 21 ((suc 𝑀 × suc 𝑀) ∈ Fin ↔ (suc 𝑀 × suc 𝑀) ≺ ω)
189187, 188sylib 208 . . . . . . . . . . . . . . . . . . . 20 (suc 𝑀 ∈ Fin → (suc 𝑀 × suc 𝑀) ≺ ω)
190185, 189syl 17 . . . . . . . . . . . . . . . . . . 19 (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ ω)
191 ssdomg 8167 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ V → (ω ⊆ 𝑎 → ω ≼ 𝑎))
19214, 191ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (ω ⊆ 𝑎 → ω ≼ 𝑎)
193 sdomdomtr 8258 . . . . . . . . . . . . . . . . . . 19 (((suc 𝑀 × suc 𝑀) ≺ ω ∧ ω ≼ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
194190, 192, 193syl2an 495 . . . . . . . . . . . . . . . . . 18 ((suc 𝑀 ∈ ω ∧ ω ⊆ 𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
195194expcom 450 . . . . . . . . . . . . . . . . 17 (ω ⊆ 𝑎 → (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
196184, 195syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
197 breq1 4807 . . . . . . . . . . . . . . . . . 18 (𝑚 = suc 𝑀 → (𝑚𝑎 ↔ suc 𝑀𝑎))
198126adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 𝑚𝑎)
199197, 198, 141rspcdva 3455 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → suc 𝑀𝑎)
200 omelon 8716 . . . . . . . . . . . . . . . . . . 19 ω ∈ On
201 ontri1 5918 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ suc 𝑀 ∈ On) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
202200, 143, 201sylancr 698 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 ↔ ¬ suc 𝑀 ∈ ω))
203 sseq2 3768 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → (ω ⊆ 𝑚 ↔ ω ⊆ suc 𝑀))
204 xpeq12 5291 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = suc 𝑀𝑚 = suc 𝑀) → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
205204anidms 680 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀 → (𝑚 × 𝑚) = (suc 𝑀 × suc 𝑀))
206 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 = suc 𝑀𝑚 = suc 𝑀)
207205, 206breq12d 4817 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = suc 𝑀 → ((𝑚 × 𝑚) ≈ 𝑚 ↔ (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
208203, 207imbi12d 333 . . . . . . . . . . . . . . . . . . 19 (𝑚 = suc 𝑀 → ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ↔ (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀)))
209 simplr 809 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
21013, 209sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
211210adantr 472 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
212208, 211, 141rspcdva 3455 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (ω ⊆ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
213202, 212sylbird 250 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (¬ suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≈ suc 𝑀))
214 ensdomtr 8261 . . . . . . . . . . . . . . . . . 18 (((suc 𝑀 × suc 𝑀) ≈ suc 𝑀 ∧ suc 𝑀𝑎) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
215214expcom 450 . . . . . . . . . . . . . . . . 17 (suc 𝑀𝑎 → ((suc 𝑀 × suc 𝑀) ≈ suc 𝑀 → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
216199, 213, 215sylsyld 61 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (¬ suc 𝑀 ∈ ω → (suc 𝑀 × suc 𝑀) ≺ 𝑎))
217196, 216pm2.61d 170 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (suc 𝑀 × suc 𝑀) ≺ 𝑎)
218 domsdomtr 8260 . . . . . . . . . . . . . . 15 (((𝑄 “ {𝑤}) ≼ (suc 𝑀 × suc 𝑀) ∧ (suc 𝑀 × suc 𝑀) ≺ 𝑎) → (𝑄 “ {𝑤}) ≺ 𝑎)
219183, 217, 218syl2anc 696 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝑄 “ {𝑤}) ≺ 𝑎)
220 ensdomtr 8261 . . . . . . . . . . . . . 14 (((𝐽𝑤) ≈ (𝑄 “ {𝑤}) ∧ (𝑄 “ {𝑤}) ≺ 𝑎) → (𝐽𝑤) ≺ 𝑎)
22188, 219, 220syl2anc 696 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ≺ 𝑎)
222 ordelon 5908 . . . . . . . . . . . . . . 15 ((Ord dom 𝐽 ∧ (𝐽𝑤) ∈ dom 𝐽) → (𝐽𝑤) ∈ On)
22376, 79, 222sylancr 698 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ On)
224 onenon 8965 . . . . . . . . . . . . . . 15 (𝑎 ∈ On → 𝑎 ∈ dom card)
225111, 224syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → 𝑎 ∈ dom card)
226 cardsdomel 8990 . . . . . . . . . . . . . 14 (((𝐽𝑤) ∈ On ∧ 𝑎 ∈ dom card) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
227223, 225, 226syl2anc 696 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ≺ 𝑎 ↔ (𝐽𝑤) ∈ (card‘𝑎)))
228221, 227mpbid 222 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ (card‘𝑎))
229 eleq2 2828 . . . . . . . . . . . . . 14 ((card‘𝑎) = 𝑎 → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
230129, 229sylbir 225 . . . . . . . . . . . . 13 ((𝑎 ∈ On ∧ ∀𝑚𝑎 𝑚𝑎) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
23117, 198, 230syl2an2r 911 . . . . . . . . . . . 12 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → ((𝐽𝑤) ∈ (card‘𝑎) ↔ (𝐽𝑤) ∈ 𝑎))
232228, 231mpbid 222 . . . . . . . . . . 11 ((𝜑𝑤 ∈ (𝑎 × 𝑎)) → (𝐽𝑤) ∈ 𝑎)
233232ralrimiva 3104 . . . . . . . . . 10 (𝜑 → ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎)
234 fnfvrnss 6553 . . . . . . . . . . 11 ((𝐽 Fn (𝑎 × 𝑎) ∧ ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎) → ran 𝐽𝑎)
235 ssdomg 8167 . . . . . . . . . . 11 (𝑎 ∈ V → (ran 𝐽𝑎 → ran 𝐽𝑎))
23614, 234, 235mpsyl 68 . . . . . . . . . 10 ((𝐽 Fn (𝑎 × 𝑎) ∧ ∀𝑤 ∈ (𝑎 × 𝑎)(𝐽𝑤) ∈ 𝑎) → ran 𝐽𝑎)
23745, 233, 236syl2anc 696 . . . . . . . . 9 (𝜑 → ran 𝐽𝑎)
238 endomtr 8179 . . . . . . . . 9 (((𝑎 × 𝑎) ≈ ran 𝐽 ∧ ran 𝐽𝑎) → (𝑎 × 𝑎) ≼ 𝑎)
23943, 237, 238syl2anc 696 . . . . . . . 8 (𝜑 → (𝑎 × 𝑎) ≼ 𝑎)
24013, 239sylbir 225 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≼ 𝑎)
241 df1o2 7741 . . . . . . . . . . . 12 1𝑜 = {∅}
242 1onn 7888 . . . . . . . . . . . 12 1𝑜 ∈ ω
243241, 242eqeltrri 2836 . . . . . . . . . . 11 {∅} ∈ ω
244 nnsdom 8724 . . . . . . . . . . 11 ({∅} ∈ ω → {∅} ≺ ω)
245 sdomdom 8149 . . . . . . . . . . 11 ({∅} ≺ ω → {∅} ≼ ω)
246243, 244, 245mp2b 10 . . . . . . . . . 10 {∅} ≼ ω
247 domtr 8174 . . . . . . . . . 10 (({∅} ≼ ω ∧ ω ≼ 𝑎) → {∅} ≼ 𝑎)
248246, 192, 247sylancr 698 . . . . . . . . 9 (ω ⊆ 𝑎 → {∅} ≼ 𝑎)
249 0ex 4942 . . . . . . . . . . . 12 ∅ ∈ V
25014, 249xpsnen 8209 . . . . . . . . . . 11 (𝑎 × {∅}) ≈ 𝑎
251250ensymi 8171 . . . . . . . . . 10 𝑎 ≈ (𝑎 × {∅})
25214xpdom2 8220 . . . . . . . . . 10 ({∅} ≼ 𝑎 → (𝑎 × {∅}) ≼ (𝑎 × 𝑎))
253 endomtr 8179 . . . . . . . . . 10 ((𝑎 ≈ (𝑎 × {∅}) ∧ (𝑎 × {∅}) ≼ (𝑎 × 𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
254251, 252, 253sylancr 698 . . . . . . . . 9 ({∅} ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎))
255248, 254syl 17 . . . . . . . 8 (ω ⊆ 𝑎𝑎 ≼ (𝑎 × 𝑎))
256255ad2antrl 766 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ≼ (𝑎 × 𝑎))
257 sbth 8245 . . . . . . 7 (((𝑎 × 𝑎) ≼ 𝑎𝑎 ≼ (𝑎 × 𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
258240, 256, 257syl2anc 696 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
259258expr 644 . . . . 5 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (∀𝑚𝑎 𝑚𝑎 → (𝑎 × 𝑎) ≈ 𝑎))
260 simplr 809 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚))
261 simpll 807 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → 𝑎 ∈ On)
262 simprr 813 . . . . . . . . 9 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ¬ ∀𝑚𝑎 𝑚𝑎)
263 rexnal 3133 . . . . . . . . . 10 (∃𝑚𝑎 ¬ 𝑚𝑎 ↔ ¬ ∀𝑚𝑎 𝑚𝑎)
264 onelss 5927 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
265 ssdomg 8167 . . . . . . . . . . . . 13 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
266264, 265syld 47 . . . . . . . . . . . 12 (𝑎 ∈ On → (𝑚𝑎𝑚𝑎))
267 bren2 8152 . . . . . . . . . . . . 13 (𝑚𝑎 ↔ (𝑚𝑎 ∧ ¬ 𝑚𝑎))
268267simplbi2 656 . . . . . . . . . . . 12 (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎))
269266, 268syl6 35 . . . . . . . . . . 11 (𝑎 ∈ On → (𝑚𝑎 → (¬ 𝑚𝑎𝑚𝑎)))
270269reximdvai 3153 . . . . . . . . . 10 (𝑎 ∈ On → (∃𝑚𝑎 ¬ 𝑚𝑎 → ∃𝑚𝑎 𝑚𝑎))
271263, 270syl5bir 233 . . . . . . . . 9 (𝑎 ∈ On → (¬ ∀𝑚𝑎 𝑚𝑎 → ∃𝑚𝑎 𝑚𝑎))
272261, 262, 271sylc 65 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∃𝑚𝑎 𝑚𝑎)
273 r19.29 3210 . . . . . . . 8 ((∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ ∃𝑚𝑎 𝑚𝑎) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
274260, 272, 273syl2anc 696 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎))
275 simprl 811 . . . . . . . 8 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → ω ⊆ 𝑎)
276 onelon 5909 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ On ∧ 𝑚𝑎) → 𝑚 ∈ On)
277 ensym 8170 . . . . . . . . . . . . . . . . . 18 (𝑚𝑎𝑎𝑚)
278 domentr 8180 . . . . . . . . . . . . . . . . . 18 ((ω ≼ 𝑎𝑎𝑚) → ω ≼ 𝑚)
279192, 277, 278syl2an 495 . . . . . . . . . . . . . . . . 17 ((ω ⊆ 𝑎𝑚𝑎) → ω ≼ 𝑚)
280 domnsym 8251 . . . . . . . . . . . . . . . . . . 19 (ω ≼ 𝑚 → ¬ 𝑚 ≺ ω)
281 nnsdom 8724 . . . . . . . . . . . . . . . . . . 19 (𝑚 ∈ ω → 𝑚 ≺ ω)
282280, 281nsyl 135 . . . . . . . . . . . . . . . . . 18 (ω ≼ 𝑚 → ¬ 𝑚 ∈ ω)
283 ontri1 5918 . . . . . . . . . . . . . . . . . . 19 ((ω ∈ On ∧ 𝑚 ∈ On) → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
284200, 283mpan 708 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → (ω ⊆ 𝑚 ↔ ¬ 𝑚 ∈ ω))
285282, 284syl5ibr 236 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ On → (ω ≼ 𝑚 → ω ⊆ 𝑚))
286276, 279, 285syl2im 40 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ On ∧ 𝑚𝑎) → ((ω ⊆ 𝑎𝑚𝑎) → ω ⊆ 𝑚))
287286expd 451 . . . . . . . . . . . . . . 15 ((𝑎 ∈ On ∧ 𝑚𝑎) → (ω ⊆ 𝑎 → (𝑚𝑎 → ω ⊆ 𝑚)))
288287impcom 445 . . . . . . . . . . . . . 14 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → (𝑚𝑎 → ω ⊆ 𝑚))
289288imim1d 82 . . . . . . . . . . . . 13 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) → (𝑚𝑎 → (𝑚 × 𝑚) ≈ 𝑚)))
290289imp32 448 . . . . . . . . . . . 12 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → (𝑚 × 𝑚) ≈ 𝑚)
291 entr 8173 . . . . . . . . . . . . . . . 16 (((𝑚 × 𝑚) ≈ 𝑚𝑚𝑎) → (𝑚 × 𝑚) ≈ 𝑎)
292291ancoms 468 . . . . . . . . . . . . . . 15 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑚 × 𝑚) ≈ 𝑎)
293 xpen 8288 . . . . . . . . . . . . . . . . 17 ((𝑎𝑚𝑎𝑚) → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
294293anidms 680 . . . . . . . . . . . . . . . 16 (𝑎𝑚 → (𝑎 × 𝑎) ≈ (𝑚 × 𝑚))
295 entr 8173 . . . . . . . . . . . . . . . 16 (((𝑎 × 𝑎) ≈ (𝑚 × 𝑚) ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
296294, 295sylan 489 . . . . . . . . . . . . . . 15 ((𝑎𝑚 ∧ (𝑚 × 𝑚) ≈ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
297277, 292, 296syl2an2r 911 . . . . . . . . . . . . . 14 ((𝑚𝑎 ∧ (𝑚 × 𝑚) ≈ 𝑚) → (𝑎 × 𝑎) ≈ 𝑎)
298297ex 449 . . . . . . . . . . . . 13 (𝑚𝑎 → ((𝑚 × 𝑚) ≈ 𝑚 → (𝑎 × 𝑎) ≈ 𝑎))
299298ad2antll 767 . . . . . . . . . . . 12 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → ((𝑚 × 𝑚) ≈ 𝑚 → (𝑎 × 𝑎) ≈ 𝑎))
300290, 299mpd 15 . . . . . . . . . . 11 (((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) ∧ ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
301300ex 449 . . . . . . . . . 10 ((ω ⊆ 𝑎 ∧ (𝑎 ∈ On ∧ 𝑚𝑎)) → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
302301expr 644 . . . . . . . . 9 ((ω ⊆ 𝑎𝑎 ∈ On) → (𝑚𝑎 → (((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎)))
303302rexlimdv 3168 . . . . . . . 8 ((ω ⊆ 𝑎𝑎 ∈ On) → (∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
304275, 261, 303syl2anc 696 . . . . . . 7 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → (∃𝑚𝑎 ((ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) ∧ 𝑚𝑎) → (𝑎 × 𝑎) ≈ 𝑎))
305274, 304mpd 15 . . . . . 6 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ¬ ∀𝑚𝑎 𝑚𝑎)) → (𝑎 × 𝑎) ≈ 𝑎)
306305expr 644 . . . . 5 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (¬ ∀𝑚𝑎 𝑚𝑎 → (𝑎 × 𝑎) ≈ 𝑎))
307259, 306pm2.61d 170 . . . 4 (((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ ω ⊆ 𝑎) → (𝑎 × 𝑎) ≈ 𝑎)
308307exp31 631 . . 3 (𝑎 ∈ On → (∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚) → (ω ⊆ 𝑎 → (𝑎 × 𝑎) ≈ 𝑎)))
3096, 12, 308tfis3 7222 . 2 (𝐴 ∈ On → (ω ⊆ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴))
310309imp 444 1 ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cun 3713  cin 3714  wss 3715  c0 4058  {csn 4321  cop 4327   class class class wbr 4804  {copab 4864   E cep 5178   Se wse 5223   We wwe 5224   × cxp 5264  ccnv 5265  dom cdm 5266  ran crn 5267  cres 5268  cima 5269  Ord word 5883  Oncon0 5884  Lim wlim 5885  suc csuc 5886   Fn wfn 6044  wf 6045  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049   Isom wiso 6050  ωcom 7230  1st c1st 7331  2nd c2nd 7332  1𝑜c1o 7722  cen 8118  cdom 8119  csdm 8120  Fincfn 8121  OrdIsocoi 8579  cardccrd 8951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-oi 8580  df-card 8955
This theorem is referenced by:  infxpen  9027
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