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Theorem ttukeylem7 9937
 Description: Lemma for ttukey 9940. (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 6676 . . . 4 (card‘( 𝐴𝐵)) ∈ V
21sucid 6259 . . 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 9936 . . 3 ((𝜑 ∧ (card‘( 𝐴𝐵)) ∈ suc (card‘( 𝐴𝐵))) → (𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴)
82, 7mpan2 690 . 2 (𝜑 → (𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴)
93, 4, 5, 6ttukeylem4 9934 . . 3 (𝜑 → (𝐺‘∅) = 𝐵)
10 0elon 6233 . . . . 5 ∅ ∈ On
11 cardon 9372 . . . . 5 (card‘( 𝐴𝐵)) ∈ On
12 0ss 4333 . . . . 5 ∅ ⊆ (card‘( 𝐴𝐵))
1310, 11, 123pm3.2i 1336 . . . 4 (∅ ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ ∅ ⊆ (card‘( 𝐴𝐵)))
143, 4, 5, 6ttukeylem5 9935 . . . 4 ((𝜑 ∧ (∅ ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ ∅ ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘∅) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
1513, 14mpan2 690 . . 3 (𝜑 → (𝐺‘∅) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
169, 15eqsstrrd 3992 . 2 (𝜑𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
17 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)
18 ssun1 4134 . . . . . . . 8 𝑦 ⊆ (𝑦𝐵)
19 undif1 4407 . . . . . . . 8 ((𝑦𝐵) ∪ 𝐵) = (𝑦𝐵)
2018, 19sseqtrri 3990 . . . . . . 7 𝑦 ⊆ ((𝑦𝐵) ∪ 𝐵)
21 simpl 486 . . . . . . . . . . . 12 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝜑)
22 f1ocnv 6620 . . . . . . . . . . . . . . . . 17 (𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) → 𝐹:( 𝐴𝐵)–1-1-onto→(card‘( 𝐴𝐵)))
23 f1of 6608 . . . . . . . . . . . . . . . . 17 (𝐹:( 𝐴𝐵)–1-1-onto→(card‘( 𝐴𝐵)) → 𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
243, 22, 233syl 18 . . . . . . . . . . . . . . . 16 (𝜑𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
2524adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝐹:( 𝐴𝐵)⟶(card‘( 𝐴𝐵)))
26 eldifi 4089 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (𝑦𝐵) → 𝑎𝑦)
2726ad2antll 728 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎𝑦)
28 simprll 778 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑦𝐴)
29 elunii 4829 . . . . . . . . . . . . . . . . 17 ((𝑎𝑦𝑦𝐴) → 𝑎 𝐴)
3027, 28, 29syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 𝐴)
31 eldifn 4090 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (𝑦𝐵) → ¬ 𝑎𝐵)
3231ad2antll 728 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ 𝑎𝐵)
3330, 32eldifd 3930 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ ( 𝐴𝐵))
3425, 33ffvelrnd 6845 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ (card‘( 𝐴𝐵)))
35 onelon 6205 . . . . . . . . . . . . . 14 (((card‘( 𝐴𝐵)) ∈ On ∧ (𝐹𝑎) ∈ (card‘( 𝐴𝐵))) → (𝐹𝑎) ∈ On)
3611, 34, 35sylancr 590 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ On)
37 suceloni 7524 . . . . . . . . . . . . 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 6284 . . . . . . . . . . . . 13 Ord (card‘( 𝐴𝐵))
41 ordsucss 7529 . . . . . . . . . . . . 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 9935 . . . . . . . . . . . 12 ((𝜑 ∧ (suc (𝐹𝑎) ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ suc (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
4421, 38, 39, 42, 43syl13anc 1369 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
45 ssun2 4135 . . . . . . . . . . . . 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 6190 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑎) ∈ On → Ord (𝐹𝑎))
47 ordunisuc 7543 . . . . . . . . . . . . . . . . . 18 (Ord (𝐹𝑎) → suc (𝐹𝑎) = (𝐹𝑎))
4836, 46, 473syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) = (𝐹𝑎))
4948fveq2d 6667 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹 suc (𝐹𝑎)) = (𝐹‘(𝐹𝑎)))
503adantr 484 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))
51 f1ocnvfv2 7028 . . . . . . . . . . . . . . . . 17 ((𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵) ∧ 𝑎 ∈ ( 𝐴𝐵)) → (𝐹‘(𝐹𝑎)) = 𝑎)
5250, 33, 51syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹‘(𝐹𝑎)) = 𝑎)
5349, 52eqtr2d 2860 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 = (𝐹 suc (𝐹𝑎)))
54 velsn 4566 . . . . . . . . . . . . . . 15 (𝑎 ∈ {(𝐹 suc (𝐹𝑎))} ↔ 𝑎 = (𝐹 suc (𝐹𝑎)))
5553, 54sylibr 237 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ {(𝐹 suc (𝐹𝑎))})
5648fveq2d 6667 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) = (𝐺‘(𝐹𝑎)))
57 ordelss 6196 . . . . . . . . . . . . . . . . . . . . 21 ((Ord (card‘( 𝐴𝐵)) ∧ (𝐹𝑎) ∈ (card‘( 𝐴𝐵))) → (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))
5840, 34, 57sylancr 590 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))
593, 4, 5, 6ttukeylem5 9935 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ ((𝐹𝑎) ∈ On ∧ (card‘( 𝐴𝐵)) ∈ On ∧ (𝐹𝑎) ⊆ (card‘( 𝐴𝐵)))) → (𝐺‘(𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
6021, 36, 39, 58, 59syl13anc 1369 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘(𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
6156, 60eqsstrd 3991 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
62 simprlr 779 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)
6361, 62sstrd 3963 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐺 suc (𝐹𝑎)) ⊆ 𝑦)
6453, 27eqeltrrd 2917 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹 suc (𝐹𝑎)) ∈ 𝑦)
6564snssd 4726 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → {(𝐹 suc (𝐹𝑎))} ⊆ 𝑦)
6663, 65unssd 4148 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ⊆ 𝑦)
673, 4, 5ttukeylem2 9932 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑦𝐴 ∧ ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ⊆ 𝑦)) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴)
6821, 28, 66, 67syl12anc 835 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴)
6968iftrued 4458 . . . . . . . . . . . . . 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 2919 . . . . . . . . . . . . 13 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ if(((𝐺 suc (𝐹𝑎)) ∪ {(𝐹 suc (𝐹𝑎))}) ∈ 𝐴, {(𝐹 suc (𝐹𝑎))}, ∅))
7145, 70sseldi 3951 . . . . . . . . . . . 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 9933 . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . 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 6258 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑎) ∈ (card‘( 𝐴𝐵)) → (𝐹𝑎) ∈ suc (𝐹𝑎))
7534, 74syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → (𝐹𝑎) ∈ suc (𝐹𝑎))
76 ordirr 6198 . . . . . . . . . . . . . . . . . 18 (Ord (𝐹𝑎) → ¬ (𝐹𝑎) ∈ (𝐹𝑎))
7736, 46, 763syl 18 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ (𝐹𝑎) ∈ (𝐹𝑎))
78 nelne1 3110 . . . . . . . . . . . . . . . . 17 (((𝐹𝑎) ∈ suc (𝐹𝑎) ∧ ¬ (𝐹𝑎) ∈ (𝐹𝑎)) → suc (𝐹𝑎) ≠ (𝐹𝑎))
7975, 77, 78syl2anc 587 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ≠ (𝐹𝑎))
8079, 48neeqtrrd 3088 . . . . . . . . . . . . . . 15 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → suc (𝐹𝑎) ≠ suc (𝐹𝑎))
8180neneqd 3019 . . . . . . . . . . . . . 14 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → ¬ suc (𝐹𝑎) = suc (𝐹𝑎))
8281iffalsed 4461 . . . . . . . . . . . . 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 2859 . . . . . . . . . . . 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 2919 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ (𝐺‘suc (𝐹𝑎)))
8544, 84sseldd 3954 . . . . . . . . . 10 ((𝜑 ∧ ((𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦𝐵))) → 𝑎 ∈ (𝐺‘(card‘( 𝐴𝐵))))
8685expr 460 . . . . . . . . 9 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝑎 ∈ (𝑦𝐵) → 𝑎 ∈ (𝐺‘(card‘( 𝐴𝐵)))))
8786ssrdv 3959 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝑦𝐵) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
8816adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → 𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
8987, 88unssd 4148 . . . . . . 7 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → ((𝑦𝐵) ∪ 𝐵) ⊆ (𝐺‘(card‘( 𝐴𝐵))))
9020, 89sstrid 3964 . . . . . 6 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → 𝑦 ⊆ (𝐺‘(card‘( 𝐴𝐵))))
9117, 90eqssd 3970 . . . . 5 ((𝜑 ∧ (𝑦𝐴 ∧ (𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘( 𝐴𝐵))) = 𝑦)
9291expr 460 . . . 4 ((𝜑𝑦𝐴) → ((𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦 → (𝐺‘(card‘( 𝐴𝐵))) = 𝑦))
93 npss 4073 . . . 4 (¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦 ↔ ((𝐺‘(card‘( 𝐴𝐵))) ⊆ 𝑦 → (𝐺‘(card‘( 𝐴𝐵))) = 𝑦))
9492, 93sylibr 237 . . 3 ((𝜑𝑦𝐴) → ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)
9594ralrimiva 3177 . 2 (𝜑 → ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)
96 sseq2 3979 . . . 4 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (𝐵𝑥𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵)))))
97 psseq1 4050 . . . . . 6 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (𝑥𝑦 ↔ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
9897notbid 321 . . . . 5 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (¬ 𝑥𝑦 ↔ ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
9998ralbidv 3192 . . . 4 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → (∀𝑦𝐴 ¬ 𝑥𝑦 ↔ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦))
10096, 99anbi12d 633 . . 3 (𝑥 = (𝐺‘(card‘( 𝐴𝐵))) → ((𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦) ↔ (𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))) ∧ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)))
101100rspcev 3609 . 2 (((𝐺‘(card‘( 𝐴𝐵))) ∈ 𝐴 ∧ (𝐵 ⊆ (𝐺‘(card‘( 𝐴𝐵))) ∧ ∀𝑦𝐴 ¬ (𝐺‘(card‘( 𝐴𝐵))) ⊊ 𝑦)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
1028, 16, 95, 101syl12anc 835 1 (𝜑 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  Vcvv 3480   ∖ cdif 3916   ∪ cun 3917   ∩ cin 3918   ⊆ wss 3919   ⊊ wpss 3920  ∅c0 4276  ifcif 4450  𝒫 cpw 4522  {csn 4550  ∪ cuni 4824   ↦ cmpt 5133  ◡ccnv 5542  dom cdm 5543  ran crn 5544   “ cima 5546  Ord word 6179  Oncon0 6180  suc csuc 6182  ⟶wf 6341  –1-1-onto→wf1o 6344  ‘cfv 6345  recscrecs 8005  Fincfn 8507  cardccrd 9363 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-om 7577  df-wrecs 7945  df-recs 8006  df-1o 8100  df-er 8287  df-en 8508  df-dom 8509  df-fin 8511  df-card 9367 This theorem is referenced by:  ttukey2g  9938
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