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Theorem ttukeylem7 9282
Description: Lemma for ttukey 9285. (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 6160 . . . 4 (card‘( 𝐴𝐵)) ∈ V
21sucid 5766 . . 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 9281 . . 3 ((𝜑 ∧ (card‘( 𝐴𝐵)) ∈ suc (card‘( 𝐴𝐵))) → (𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴)
82, 7mpan2 706 . 2 (𝜑 → (𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴)
93, 4, 5, 6ttukeylem4 9279 . . 3 (𝜑 → (𝐺‘∅) = 𝐵)
10 0elon 5740 . . . . 5 ∅ ∈ On
11 cardon 8715 . . . . 5 (card‘( 𝐴𝐵)) ∈ On
12 0ss 3949 . . . . 5 ∅ ⊆ (card‘( 𝐴𝐵))
1310, 11, 123pm3.2i 1237 . . . 4 (∅ ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ ∅ ⊆ (card‘( 𝐴𝐵)))
143, 4, 5, 6ttukeylem5 9280 . . . 4 ((𝜑 ∧ (∅ ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ ∅ ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘∅) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
1513, 14mpan2 706 . . 3 (𝜑 → (𝐺‘∅) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
169, 15eqsstr3d 3624 . 2 (𝜑𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
17 simprr 795 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)
18 ssun1 3759 . . . . . . . 8 𝑦 ⊆ (𝑦𝐵)
19 undif1 4020 . . . . . . . 8 ((𝑦𝐵) ∪ 𝐵) = (𝑦𝐵)
2018, 19sseqtr4i 3622 . . . . . . 7 𝑦 ⊆ ((𝑦𝐵) ∪ 𝐵)
21 simpl 473 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝜑)
22 f1ocnv 6108 . . . . . . . . . . . . . . . . 17 (𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → 𝐹:( 𝐴𝐵)–1-1-onto→(card‘( 𝐴𝐵)))
23 f1of 6096 . . . . . . . . . . . . . . . . 17 (𝐹:( 𝐴𝐵)–1-1-onto→(card‘( 𝐴𝐵)) → 𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
243, 22, 233syl 18 . . . . . . . . . . . . . . . 16 (𝜑𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
2524adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
26 eldifi 3715 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (𝑦𝐵) → 𝑎𝑦)
2726ad2antll 764 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎𝑦)
28 simprll 801 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑦𝐴)
29 elunii 4412 . . . . . . . . . . . . . . . . 17 ((𝑎𝑦𝑦𝐴) → 𝑎 𝐴)
3027, 28, 29syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 𝐴)
31 eldifn 3716 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (𝑦𝐵) → ¬ 𝑎𝐵)
3231ad2antll 764 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ 𝑎𝐵)
3330, 32eldifd 3571 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ ( 𝐴𝐵))
3425, 33ffvelrnd 6317 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ (card‘( 𝐴𝐵)))
35 onelon 5710 . . . . . . . . . . . . . 14 (((card‘( 𝐴𝐵)) ∈ On ∧ (𝐹𝑎) ∈ (card‘( 𝐴𝐵))) → (𝐹𝑎) ∈ On)
3611, 34, 35sylancr 694 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ On)
37 suceloni 6961 . . . . . . . . . . . . 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 5794 . . . . . . . . . . . . 13 Ord (card‘( 𝐴𝐵))
41 ordsucss 6966 . . . . . . . . . . . . 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 9280 . . . . . . . . . . . 12 ((𝜑 ∧ (suc (𝐹𝑎) ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ suc (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
4421, 38, 39, 42, 43syl13anc 1325 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
45 ssun2 3760 . . . . . . . . . . . . 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 5695 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑎) ∈ On → Ord (𝐹𝑎))
47 ordunisuc 6980 . . . . . . . . . . . . . . . . . 18 (Ord (𝐹𝑎) → suc (𝐹𝑎) = (𝐹𝑎))
4836, 46, 473syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) = (𝐹𝑎))
4948fveq2d 6154 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹 suc (𝐹𝑎)) = (𝐹‘(𝐹𝑎)))
503adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
51 f1ocnvfv2 6488 . . . . . . . . . . . . . . . . 17 ((𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝑎 ∈ ( 𝐴𝐵)) → (𝐹‘(𝐹𝑎)) = 𝑎)
5250, 33, 51syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹‘(𝐹𝑎)) = 𝑎)
5349, 52eqtr2d 2661 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 = (𝐹 suc (𝐹𝑎)))
54 velsn 4169 . . . . . . . . . . . . . . 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 6154 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) = (𝐺‘(𝐹𝑎)))
57 ordelss 5701 . . . . . . . . . . . . . . . . . . . . 21 ((Ord (card‘( 𝐴𝐵)) ∧ (𝐹𝑎) ∈ (card‘( 𝐴𝐵))) → (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))
5840, 34, 57sylancr 694 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))
593, 4, 5, 6ttukeylem5 9280 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝐹𝑎) ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘(𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
6021, 36, 39, 58, 59syl13anc 1325 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘(𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
6156, 60eqsstrd 3623 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
62 simprlr 802 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)
6361, 62sstrd 3598 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) ⊆ 𝑦)
6453, 27eqeltrrd 2705 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹 suc (𝐹𝑎)) ∈ 𝑦)
6564snssd 4314 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → {(𝐹 suc (𝐹𝑎))} ⊆ 𝑦)
6663, 65unssd 3772 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ⊆ 𝑦)
673, 4, 5ttukeylem2 9277 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝐴 ∧ ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ⊆ 𝑦)) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴)
6821, 28, 66, 67syl12anc 1321 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴)
6968iftrued 4071 . . . . . . . . . . . . . 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 2707 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅))
7145, 70sseldi 3586 . . . . . . . . . . . 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 9278 . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . 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 5765 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑎) ∈ (card‘( 𝐴𝐵)) → (𝐹𝑎) ∈ suc (𝐹𝑎))
7534, 74syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ suc (𝐹𝑎))
76 ordirr 5703 . . . . . . . . . . . . . . . . . 18 (Ord (𝐹𝑎) → ¬ (𝐹𝑎) ∈ (𝐹𝑎))
7736, 46, 763syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ (𝐹𝑎) ∈ (𝐹𝑎))
78 nelne1 2892 . . . . . . . . . . . . . . . . 17 (((𝐹𝑎) ∈ suc (𝐹𝑎) ∧ ¬ (𝐹𝑎) ∈ (𝐹𝑎)) → suc (𝐹𝑎) ≠ (𝐹𝑎))
7975, 77, 78syl2anc 692 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ≠ (𝐹𝑎))
8079, 48neeqtrrd 2870 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ≠ suc (𝐹𝑎))
8180neneqd 2801 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ suc (𝐹𝑎) = suc (𝐹𝑎))
8281iffalsed 4074 . . . . . . . . . . . . 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 2660 . . . . . . . . . . . 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 2707 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ (𝐺‘suc (𝐹𝑎)))
8544, 84sseldd 3589 . . . . . . . . . 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 3594 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝑦𝐵) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
8816adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → 𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
8987, 88unssd 3772 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → ((𝑦𝐵) ∪ 𝐵) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
9020, 89syl5ss 3599 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → 𝑦 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
9117, 90eqssd 3605 . . . . 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 3700 . . . 4 (¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦 ↔ ((𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦 → (𝐺‘(card‘( 𝐴𝐵))) = 𝑦))
9492, 93sylibr 224 . . 3 ((𝜑𝑦𝐴) → ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)
9594ralrimiva 2965 . 2 (𝜑 → ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)
96 sseq2 3611 . . . 4 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (𝐵𝑥𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵)))))
97 psseq1 3677 . . . . . 6 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (𝑥𝑦 ↔ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
9897notbid 308 . . . . 5 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (¬ 𝑥𝑦 ↔ ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
9998ralbidv 2985 . . . 4 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
10096, 99anbi12d 746 . . 3 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → ((𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦) ↔ (𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))) ∧ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)))
101100rspcev 3300 . 2 (((𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴 ∧ (𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))) ∧ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
1028, 16, 95, 101syl12anc 1321 1 (𝜑 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1992  wne 2796  wral 2912  wrex 2913  Vcvv 3191  cdif 3557  cun 3558  cin 3559  wss 3560  wpss 3561  c0 3896  ifcif 4063  𝒫 cpw 4135  {csn 4153   cuni 4407  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  Ord word 5684  Oncon0 5685  suc csuc 5687  wf 5846  1-1-ontowf1o 5849  cfv 5850  recscrecs 7413  Fincfn 7900  cardccrd 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-wrecs 7353  df-recs 7414  df-1o 7506  df-er 7688  df-en 7901  df-dom 7902  df-fin 7904  df-card 8710
This theorem is referenced by:  ttukey2g  9283
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