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Theorem ttukeylem7 9375
Description: Lemma for ttukey 9378. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
ttukeylem.1 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
ttukeylem.2 (𝜑𝐵𝐴)
ttukeylem.3 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
ttukeylem.4 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
Assertion
Ref Expression
ttukeylem7 (𝜑 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐺   𝜑,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑧
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑦)

Proof of Theorem ttukeylem7
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fvex 6239 . . . 4 (card‘( 𝐴𝐵)) ∈ V
21sucid 5842 . . 3 (card‘( 𝐴𝐵)) ∈ suc (card‘( 𝐴𝐵))
3 ttukeylem.1 . . . 4 (𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
4 ttukeylem.2 . . . 4 (𝜑𝐵𝐴)
5 ttukeylem.3 . . . 4 (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))
6 ttukeylem.4 . . . 4 𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))
73, 4, 5, 6ttukeylem6 9374 . . 3 ((𝜑 ∧ (card‘( 𝐴𝐵)) ∈ suc (card‘( 𝐴𝐵))) → (𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴)
82, 7mpan2 707 . 2 (𝜑 → (𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴)
93, 4, 5, 6ttukeylem4 9372 . . 3 (𝜑 → (𝐺‘∅) = 𝐵)
10 0elon 5816 . . . . 5 ∅ ∈ On
11 cardon 8808 . . . . 5 (card‘( 𝐴𝐵)) ∈ On
12 0ss 4005 . . . . 5 ∅ ⊆ (card‘( 𝐴𝐵))
1310, 11, 123pm3.2i 1259 . . . 4 (∅ ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ ∅ ⊆ (card‘( 𝐴𝐵)))
143, 4, 5, 6ttukeylem5 9373 . . . 4 ((𝜑 ∧ (∅ ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ ∅ ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘∅) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
1513, 14mpan2 707 . . 3 (𝜑 → (𝐺‘∅) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
169, 15eqsstr3d 3673 . 2 (𝜑𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
17 simprr 811 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)
18 ssun1 3809 . . . . . . . 8 𝑦 ⊆ (𝑦𝐵)
19 undif1 4076 . . . . . . . 8 ((𝑦𝐵) ∪ 𝐵) = (𝑦𝐵)
2018, 19sseqtr4i 3671 . . . . . . 7 𝑦 ⊆ ((𝑦𝐵) ∪ 𝐵)
21 simpl 472 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝜑)
22 f1ocnv 6187 . . . . . . . . . . . . . . . . 17 (𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → 𝐹:( 𝐴𝐵)–1-1-onto→(card‘( 𝐴𝐵)))
23 f1of 6175 . . . . . . . . . . . . . . . . 17 (𝐹:( 𝐴𝐵)–1-1-onto→(card‘( 𝐴𝐵)) → 𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
243, 22, 233syl 18 . . . . . . . . . . . . . . . 16 (𝜑𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
2524adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
26 eldifi 3765 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (𝑦𝐵) → 𝑎𝑦)
2726ad2antll 765 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎𝑦)
28 simprll 819 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑦𝐴)
29 elunii 4473 . . . . . . . . . . . . . . . . 17 ((𝑎𝑦𝑦𝐴) → 𝑎 𝐴)
3027, 28, 29syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 𝐴)
31 eldifn 3766 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (𝑦𝐵) → ¬ 𝑎𝐵)
3231ad2antll 765 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ 𝑎𝐵)
3330, 32eldifd 3618 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ ( 𝐴𝐵))
3425, 33ffvelrnd 6400 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ (card‘( 𝐴𝐵)))
35 onelon 5786 . . . . . . . . . . . . . 14 (((card‘( 𝐴𝐵)) ∈ On ∧ (𝐹𝑎) ∈ (card‘( 𝐴𝐵))) → (𝐹𝑎) ∈ On)
3611, 34, 35sylancr 696 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ On)
37 suceloni 7055 . . . . . . . . . . . . 13 ((𝐹𝑎) ∈ On → suc (𝐹𝑎) ∈ On)
3836, 37syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ∈ On)
3911a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (card‘( 𝐴𝐵)) ∈ On)
4011onordi 5870 . . . . . . . . . . . . 13 Ord (card‘( 𝐴𝐵))
41 ordsucss 7060 . . . . . . . . . . . . 13 (Ord (card‘( 𝐴𝐵)) → ((𝐹𝑎) ∈ (card‘( 𝐴𝐵)) → suc (𝐹𝑎) ⊆ (card‘( 𝐴𝐵))))
4240, 34, 41mpsyl 68 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))
433, 4, 5, 6ttukeylem5 9373 . . . . . . . . . . . 12 ((𝜑 ∧ (suc (𝐹𝑎) ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ suc (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
4421, 38, 39, 42, 43syl13anc 1368 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
45 ssun2 3810 . . . . . . . . . . . . 13 if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅) ⊆ ((𝐺 suc (𝐹𝑎)) ∪ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅))
46 eloni 5771 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑎) ∈ On → Ord (𝐹𝑎))
47 ordunisuc 7074 . . . . . . . . . . . . . . . . . 18 (Ord (𝐹𝑎) → suc (𝐹𝑎) = (𝐹𝑎))
4836, 46, 473syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) = (𝐹𝑎))
4948fveq2d 6233 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹 suc (𝐹𝑎)) = (𝐹‘(𝐹𝑎)))
503adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
51 f1ocnvfv2 6573 . . . . . . . . . . . . . . . . 17 ((𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝑎 ∈ ( 𝐴𝐵)) → (𝐹‘(𝐹𝑎)) = 𝑎)
5250, 33, 51syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹‘(𝐹𝑎)) = 𝑎)
5349, 52eqtr2d 2686 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 = (𝐹 suc (𝐹𝑎)))
54 velsn 4226 . . . . . . . . . . . . . . 15 (𝑎 ∈ {(𝐹 suc (𝐹𝑎))} ↔ 𝑎 = (𝐹 suc (𝐹𝑎)))
5553, 54sylibr 224 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ {(𝐹 suc (𝐹𝑎))})
5648fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) = (𝐺‘(𝐹𝑎)))
57 ordelss 5777 . . . . . . . . . . . . . . . . . . . . 21 ((Ord (card‘( 𝐴𝐵)) ∧ (𝐹𝑎) ∈ (card‘( 𝐴𝐵))) → (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))
5840, 34, 57sylancr 696 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))
593, 4, 5, 6ttukeylem5 9373 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝐹𝑎) ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘(𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
6021, 36, 39, 58, 59syl13anc 1368 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘(𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
6156, 60eqsstrd 3672 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
62 simprlr 820 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)
6361, 62sstrd 3646 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) ⊆ 𝑦)
6453, 27eqeltrrd 2731 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹 suc (𝐹𝑎)) ∈ 𝑦)
6564snssd 4372 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → {(𝐹 suc (𝐹𝑎))} ⊆ 𝑦)
6663, 65unssd 3822 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ⊆ 𝑦)
673, 4, 5ttukeylem2 9370 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝐴 ∧ ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ⊆ 𝑦)) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴)
6821, 28, 66, 67syl12anc 1364 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴)
6968iftrued 4127 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅) = {(𝐹 suc (𝐹𝑎))})
7055, 69eleqtrrd 2733 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅))
7145, 70sseldi 3634 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ ((𝐺 suc (𝐹𝑎)) ∪ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅)))
723, 4, 5, 6ttukeylem3 9371 . . . . . . . . . . . . . 14 ((𝜑 ∧ suc (𝐹𝑎) ∈ On) → (𝐺‘suc (𝐹𝑎)) = if(suc (𝐹𝑎) = suc (𝐹𝑎), if(suc (𝐹𝑎) = ∅, 𝐵, (𝐺 “ suc (𝐹𝑎))), ((𝐺 suc (𝐹𝑎)) ∪ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅))))
7338, 72syldan 486 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘suc (𝐹𝑎)) = if(suc (𝐹𝑎) = suc (𝐹𝑎), if(suc (𝐹𝑎) = ∅, 𝐵, (𝐺 “ suc (𝐹𝑎))), ((𝐺 suc (𝐹𝑎)) ∪ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅))))
74 sucidg 5841 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑎) ∈ (card‘( 𝐴𝐵)) → (𝐹𝑎) ∈ suc (𝐹𝑎))
7534, 74syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ suc (𝐹𝑎))
76 ordirr 5779 . . . . . . . . . . . . . . . . . 18 (Ord (𝐹𝑎) → ¬ (𝐹𝑎) ∈ (𝐹𝑎))
7736, 46, 763syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ (𝐹𝑎) ∈ (𝐹𝑎))
78 nelne1 2919 . . . . . . . . . . . . . . . . 17 (((𝐹𝑎) ∈ suc (𝐹𝑎) ∧ ¬ (𝐹𝑎) ∈ (𝐹𝑎)) → suc (𝐹𝑎) ≠ (𝐹𝑎))
7975, 77, 78syl2anc 694 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ≠ (𝐹𝑎))
8079, 48neeqtrrd 2897 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ≠ suc (𝐹𝑎))
8180neneqd 2828 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ suc (𝐹𝑎) = suc (𝐹𝑎))
8281iffalsed 4130 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → if(suc (𝐹𝑎) = suc (𝐹𝑎), if(suc (𝐹𝑎) = ∅, 𝐵, (𝐺 “ suc (𝐹𝑎))), ((𝐺 suc (𝐹𝑎)) ∪ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅))) = ((𝐺 suc (𝐹𝑎)) ∪ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅)))
8373, 82eqtrd 2685 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘suc (𝐹𝑎)) = ((𝐺 suc (𝐹𝑎)) ∪ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅)))
8471, 83eleqtrrd 2733 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ (𝐺‘suc (𝐹𝑎)))
8544, 84sseldd 3637 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ (𝐺‘(card‘( 𝐴𝐵))))
8685expr 642 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝑎 ∈ (𝑦𝐵) → 𝑎 ∈ (𝐺‘(card‘( 𝐴𝐵)))))
8786ssrdv 3642 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝑦𝐵) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
8816adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → 𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
8987, 88unssd 3822 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → ((𝑦𝐵) ∪ 𝐵) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
9020, 89syl5ss 3647 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → 𝑦 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
9117, 90eqssd 3653 . . . . 5 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘( 𝐴𝐵))) = 𝑦)
9291expr 642 . . . 4 ((𝜑𝑦𝐴) → ((𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦 → (𝐺‘(card‘( 𝐴𝐵))) = 𝑦))
93 npss 3750 . . . 4 (¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦 ↔ ((𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦 → (𝐺‘(card‘( 𝐴𝐵))) = 𝑦))
9492, 93sylibr 224 . . 3 ((𝜑𝑦𝐴) → ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)
9594ralrimiva 2995 . 2 (𝜑 → ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)
96 sseq2 3660 . . . 4 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (𝐵𝑥𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵)))))
97 psseq1 3727 . . . . . 6 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (𝑥𝑦 ↔ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
9897notbid 307 . . . . 5 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (¬ 𝑥𝑦 ↔ ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
9998ralbidv 3015 . . . 4 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
10096, 99anbi12d 747 . . 3 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → ((𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦) ↔ (𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))) ∧ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)))
101100rspcev 3340 . 2 (((𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴 ∧ (𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))) ∧ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
1028, 16, 95, 101syl12anc 1364 1 (𝜑 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  wpss 3608  c0 3948  ifcif 4119  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  Ord word 5760  Oncon0 5761  suc csuc 5763  wf 5922  1-1-ontowf1o 5925  cfv 5926  recscrecs 7512  Fincfn 7997  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001  df-card 8803
This theorem is referenced by:  ttukey2g  9376
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