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Theorem unxpdomlem3 8323
Description: Lemma for unxpdom 8324. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
Hypotheses
Ref Expression
unxpdomlem1.1 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
unxpdomlem1.2 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
Assertion
Ref Expression
unxpdomlem3 ((1𝑜𝑎 ∧ 1𝑜𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
Distinct variable group:   𝑎,𝑏,𝑚,𝑛,𝑠,𝑡,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)   𝐺(𝑥,𝑡,𝑚,𝑛,𝑠,𝑎,𝑏)

Proof of Theorem unxpdomlem3
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3335 . . 3 𝑎 ∈ V
2 1sdom 8320 . . 3 (𝑎 ∈ V → (1𝑜𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛))
31, 2ax-mp 5 . 2 (1𝑜𝑎 ↔ ∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛)
4 vex 3335 . . 3 𝑏 ∈ V
5 1sdom 8320 . . 3 (𝑏 ∈ V → (1𝑜𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
64, 5ax-mp 5 . 2 (1𝑜𝑏 ↔ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡)
7 reeanv 3237 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) ↔ (∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡))
8 reeanv 3237 . . . . 5 (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ↔ (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡))
9 unxpdomlem1.2 . . . . . . . . . . 11 𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)
10 simpr 479 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → 𝑥𝑎)
11 simp2r 1240 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑡𝑏)
12 simp1r 1238 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑠𝑏)
1311, 12ifcld 4267 . . . . . . . . . . . . . 14 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
1413ad2antrr 764 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏)
15 opelxpi 5297 . . . . . . . . . . . . 13 ((𝑥𝑎 ∧ if(𝑥 = 𝑚, 𝑡, 𝑠) ∈ 𝑏) → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ ∈ (𝑎 × 𝑏))
1610, 14, 15syl2anc 696 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ 𝑥𝑎) → ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩ ∈ (𝑎 × 𝑏))
17 simp2l 1239 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑛𝑎)
18 simp1l 1237 . . . . . . . . . . . . . . 15 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝑚𝑎)
1917, 18ifcld 4267 . . . . . . . . . . . . . 14 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
2019ad2antrr 764 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎)
21 simpr 479 . . . . . . . . . . . . . . 15 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝑥 ∈ (𝑎𝑏))
22 elun 3888 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝑎𝑏) ↔ (𝑥𝑎𝑥𝑏))
2321, 22sylib 208 . . . . . . . . . . . . . 14 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → (𝑥𝑎𝑥𝑏))
2423orcanai 990 . . . . . . . . . . . . 13 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → 𝑥𝑏)
25 opelxpi 5297 . . . . . . . . . . . . 13 ((if(𝑥 = 𝑡, 𝑛, 𝑚) ∈ 𝑎𝑥𝑏) → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ ∈ (𝑎 × 𝑏))
2620, 24, 25syl2anc 696 . . . . . . . . . . . 12 (((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) ∧ ¬ 𝑥𝑎) → ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩ ∈ (𝑎 × 𝑏))
2716, 26ifclda 4256 . . . . . . . . . . 11 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩) ∈ (𝑎 × 𝑏))
289, 27syl5eqel 2835 . . . . . . . . . 10 ((((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) ∧ 𝑥 ∈ (𝑎𝑏)) → 𝐺 ∈ (𝑎 × 𝑏))
29 unxpdomlem1.1 . . . . . . . . . 10 𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)
3028, 29fmptd 6540 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏))
3129, 9unxpdomlem1 8321 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
3231ad2antrl 766 . . . . . . . . . . . . . . 15 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))
33 iftrue 4228 . . . . . . . . . . . . . . . 16 (𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3433adantr 472 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3532, 34sylan9eq 2806 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑧) = ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩)
3629, 9unxpdomlem1 8321 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (𝑎𝑏) → (𝐹𝑤) = if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
3736ad2antll 767 . . . . . . . . . . . . . . 15 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → (𝐹𝑤) = if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
38 iftrue 4228 . . . . . . . . . . . . . . . 16 (𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
3938adantl 473 . . . . . . . . . . . . . . 15 ((𝑧𝑎𝑤𝑎) → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4037, 39sylan9eq 2806 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → (𝐹𝑤) = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩)
4135, 40eqeq12d 2767 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩))
42 vex 3335 . . . . . . . . . . . . . 14 𝑧 ∈ V
43 vex 3335 . . . . . . . . . . . . . . 15 𝑡 ∈ V
44 vex 3335 . . . . . . . . . . . . . . 15 𝑠 ∈ V
4543, 44ifex 4292 . . . . . . . . . . . . . 14 if(𝑧 = 𝑚, 𝑡, 𝑠) ∈ V
4642, 45opth1 5084 . . . . . . . . . . . . 13 (⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩ = ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩ → 𝑧 = 𝑤)
4741, 46syl6bi 243 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
48 simprr 813 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → 𝑤 ∈ (𝑎𝑏))
49 simpll 807 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ¬ 𝑚 = 𝑛)
50 simplr 809 . . . . . . . . . . . . . 14 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ¬ 𝑠 = 𝑡)
5129, 9, 48, 49, 50unxpdomlem2 8322 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))
5251pm2.21d 118 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
53 eqcom 2759 . . . . . . . . . . . . 13 ((𝐹𝑧) = (𝐹𝑤) ↔ (𝐹𝑤) = (𝐹𝑧))
54 simprl 811 . . . . . . . . . . . . . . . 16 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → 𝑧 ∈ (𝑎𝑏))
5529, 9, 54, 49, 50unxpdomlem2 8322 . . . . . . . . . . . . . . 15 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (𝑤𝑎 ∧ ¬ 𝑧𝑎)) → ¬ (𝐹𝑤) = (𝐹𝑧))
5655ancom2s 879 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ¬ (𝐹𝑤) = (𝐹𝑧))
5756pm2.21d 118 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ((𝐹𝑤) = (𝐹𝑧) → 𝑧 = 𝑤))
5853, 57syl5bi 232 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
59 iffalse 4231 . . . . . . . . . . . . . . . 16 𝑧𝑎 → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
6059adantr 472 . . . . . . . . . . . . . . 15 ((¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎) → if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
6132, 60sylan9eq 2806 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑧) = ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩)
62 iffalse 4231 . . . . . . . . . . . . . . . 16 𝑤𝑎 → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6362adantl 473 . . . . . . . . . . . . . . 15 ((¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎) → if(𝑤𝑎, ⟨𝑤, if(𝑤 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6437, 63sylan9eq 2806 . . . . . . . . . . . . . 14 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → (𝐹𝑤) = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩)
6561, 64eqeq12d 2767 . . . . . . . . . . . . 13 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩))
66 vex 3335 . . . . . . . . . . . . . . . 16 𝑛 ∈ V
67 vex 3335 . . . . . . . . . . . . . . . 16 𝑚 ∈ V
6866, 67ifex 4292 . . . . . . . . . . . . . . 15 if(𝑧 = 𝑡, 𝑛, 𝑚) ∈ V
6968, 42opth 5085 . . . . . . . . . . . . . 14 (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ ↔ (if(𝑧 = 𝑡, 𝑛, 𝑚) = if(𝑤 = 𝑡, 𝑛, 𝑚) ∧ 𝑧 = 𝑤))
7069simprbi 483 . . . . . . . . . . . . 13 (⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩ = ⟨if(𝑤 = 𝑡, 𝑛, 𝑚), 𝑤⟩ → 𝑧 = 𝑤)
7165, 70syl6bi 243 . . . . . . . . . . . 12 ((((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) ∧ (¬ 𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7247, 52, 58, 714casesdan 1028 . . . . . . . . . . 11 (((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) ∧ (𝑧 ∈ (𝑎𝑏) ∧ 𝑤 ∈ (𝑎𝑏))) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7372ralrimivva 3101 . . . . . . . . . 10 ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
74733ad2ant3 1129 . . . . . . . . 9 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
75 dff13 6667 . . . . . . . . 9 (𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏) ↔ (𝐹:(𝑎𝑏)⟶(𝑎 × 𝑏) ∧ ∀𝑧 ∈ (𝑎𝑏)∀𝑤 ∈ (𝑎𝑏)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
7630, 74, 75sylanbrc 701 . . . . . . . 8 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏))
771, 4unex 7113 . . . . . . . . 9 (𝑎𝑏) ∈ V
781, 4xpex 7119 . . . . . . . . 9 (𝑎 × 𝑏) ∈ V
79 f1dom2g 8131 . . . . . . . . 9 (((𝑎𝑏) ∈ V ∧ (𝑎 × 𝑏) ∈ V ∧ 𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
8077, 78, 79mp3an12 1555 . . . . . . . 8 (𝐹:(𝑎𝑏)–1-1→(𝑎 × 𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
8176, 80syl 17 . . . . . . 7 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏) ∧ (¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡)) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
82813expia 1114 . . . . . 6 (((𝑚𝑎𝑠𝑏) ∧ (𝑛𝑎𝑡𝑏)) → ((¬ 𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8382rexlimdvva 3168 . . . . 5 ((𝑚𝑎𝑠𝑏) → (∃𝑛𝑎𝑡𝑏𝑚 = 𝑛 ∧ ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
848, 83syl5bir 233 . . . 4 ((𝑚𝑎𝑠𝑏) → ((∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏)))
8584rexlimivv 3166 . . 3 (∃𝑚𝑎𝑠𝑏 (∃𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
867, 85sylbir 225 . 2 ((∃𝑚𝑎𝑛𝑎 ¬ 𝑚 = 𝑛 ∧ ∃𝑠𝑏𝑡𝑏 ¬ 𝑠 = 𝑡) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
873, 6, 86syl2anb 497 1 ((1𝑜𝑎 ∧ 1𝑜𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042  wrex 3043  Vcvv 3332  cun 3705  ifcif 4222  cop 4319   class class class wbr 4796  cmpt 4873   × cxp 5256  wf 6037  1-1wf1 6038  cfv 6041  1𝑜c1o 7714  cdom 8111  csdm 8112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-1o 7721  df-2o 7722  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116
This theorem is referenced by:  unxpdom  8324
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