Theorem ttukeylem7 10487

Description: Lemma for ttukey 10490. (Contributed by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
ttukeylem.1	⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
ttukeylem.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
ttukeylem.3	⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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 → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))

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.

Step	Hyp	Ref	Expression
1		fvex 6884	. . . 4 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
2	1	sucid 6434	. . 3 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ suc (card‘(∪ 𝐴 ∖ 𝐵))
3		ttukeylem.1	. . . 4 ⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
4		ttukeylem.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
5		ttukeylem.3	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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)}, ∅)))))
7	3, 4, 5, 6	ttukeylem6 10486	. . 3 ⊢ ((𝜑 ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ suc (card‘(∪ 𝐴 ∖ 𝐵))) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∈ 𝐴)
8	2, 7	mpan2 703	. 2 ⊢ (𝜑 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∈ 𝐴)
9	3, 4, 5, 6	ttukeylem4 10484	. . 3 ⊢ (𝜑 → (𝐺‘∅) = 𝐵)
10		0elon 6405	. . . . 5 ⊢ ∅ ∈ On
11		cardon 9918	. . . . 5 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On
12		0ss 4357	. . . . 5 ⊢ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵))
13	10, 11, 12	3pm3.2i 1356	. . . 4 ⊢ (∅ ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
14	3, 4, 5, 6	ttukeylem5 10485	. . . 4 ⊢ ((𝜑 ∧ (∅ ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))) → (𝐺‘∅) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
15	13, 14	mpan2 703	. . 3 ⊢ (𝜑 → (𝐺‘∅) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
16	9, 15	eqsstrrd 3974	. 2 ⊢ (𝜑 → 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
17		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)
18		ssun1 4133	. . . . . . . 8 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝐵)
19		undif1 4433	. . . . . . . 8 ⊢ ((𝑦 ∖ 𝐵) ∪ 𝐵) = (𝑦 ∪ 𝐵)
20	18, 19	sseqtrri 3988	. . . . . . 7 ⊢ 𝑦 ⊆ ((𝑦 ∖ 𝐵) ∪ 𝐵)
21		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝜑)
22		f1ocnv 6823	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)–1-1-onto→(card‘(∪ 𝐴 ∖ 𝐵)))
23		f1of 6810	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝐹:(∪ 𝐴 ∖ 𝐵)–1-1-onto→(card‘(∪ 𝐴 ∖ 𝐵)) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)⟶(card‘(∪ 𝐴 ∖ 𝐵)))
24	3, 22, 23	3syl 19	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ◡𝐹:(∪ 𝐴 ∖ 𝐵)⟶(card‘(∪ 𝐴 ∖ 𝐵)))
25	24	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)⟶(card‘(∪ 𝐴 ∖ 𝐵)))
26		eldifi 4087	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑦 ∖ 𝐵) → 𝑎 ∈ 𝑦)
27	26	ad2antll 741	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ 𝑦)
28		simprll 790	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑦 ∈ 𝐴)
29		elunii 4873	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑎 ∈ ∪ 𝐴)
30	27, 28, 29	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ ∪ 𝐴)
31		eldifn 4088	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝑦 ∖ 𝐵) → ¬ 𝑎 ∈ 𝐵)
32	31	ad2antll 741	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ 𝑎 ∈ 𝐵)
33	30, 32	eldifd 3918	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (∪ 𝐴 ∖ 𝐵))
34	25, 33	ffvelcdmd 7070	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)))
35		onelon 6375	. . . . . . . . . . . . . 14 ⊢ (((card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ On)
36	11, 34, 35	sylancr 598	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ On)
37		onsuc 7797	. . . . . . . . . . . . 13 ⊢ ((◡𝐹‘𝑎) ∈ On → suc (◡𝐹‘𝑎) ∈ On)
38	36, 37	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ∈ On)
39	11	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On)
40	11	onordi 6463	. . . . . . . . . . . . 13 ⊢ Ord (card‘(∪ 𝐴 ∖ 𝐵))
41		ordsucss 7802	. . . . . . . . . . . . 13 ⊢ (Ord (card‘(∪ 𝐴 ∖ 𝐵)) → ((◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)) → suc (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵))))
42	40, 34, 41	mpsyl 69	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
43	3, 4, 5, 6	ttukeylem5 10485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (suc (◡𝐹‘𝑎) ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ suc (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))) → (𝐺‘suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
44	21, 38, 39, 42, 43	syl13anc 1395	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
45		ssun2 4134	. . . . . . . . . . . . 13 ⊢ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅) ⊆ ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅))
46		eloni 6360	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝐹‘𝑎) ∈ On → Ord (◡𝐹‘𝑎))
47		ordunisuc 7816	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord (◡𝐹‘𝑎) → ∪ suc (◡𝐹‘𝑎) = (◡𝐹‘𝑎))
48	36, 46, 47	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ∪ suc (◡𝐹‘𝑎) = (◡𝐹‘𝑎))
49	48	fveq2d 6875	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘∪ suc (◡𝐹‘𝑎)) = (𝐹‘(◡𝐹‘𝑎)))
50	3	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
51		f1ocnvfv2 7265	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝑎 ∈ (∪ 𝐴 ∖ 𝐵)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
52	50, 33, 51	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
53	49, 52	eqtr2d 2801	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 = (𝐹‘∪ suc (◡𝐹‘𝑎)))
54		velsn 4601	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {(𝐹‘∪ suc (◡𝐹‘𝑎))} ↔ 𝑎 = (𝐹‘∪ suc (◡𝐹‘𝑎)))
55	53, 54	sylibr 237	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ {(𝐹‘∪ suc (◡𝐹‘𝑎))})
56	48	fveq2d 6875	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) = (𝐺‘(◡𝐹‘𝑎)))
57		ordelss 6366	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Ord (card‘(∪ 𝐴 ∖ 𝐵)) ∧ (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵))) → (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
58	40, 34, 57	sylancr 598	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
59	3, 4, 5, 6	ttukeylem5 10485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((◡𝐹‘𝑎) ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))) → (𝐺‘(◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
60	21, 36, 39, 58, 59	syl13anc 1395	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘(◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
61	56, 60	eqsstrd 3973	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
62		simprlr 791	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)
63	61, 62	sstrd 3949	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) ⊆ 𝑦)
64	53, 27	eqeltrrd 2866	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘∪ suc (◡𝐹‘𝑎)) ∈ 𝑦)
65	64	snssd 4748	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → {(𝐹‘∪ suc (◡𝐹‘𝑎))} ⊆ 𝑦)
66	63, 65	unssd 4147	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ⊆ 𝑦)
67	3, 4, 5	ttukeylem2 10482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ⊆ 𝑦)) → ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴)
68	21, 28, 66, 67	syl12anc 849	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴)
69	68	iftrued 4491	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅) = {(𝐹‘∪ suc (◡𝐹‘𝑎))})
70	55, 69	eleqtrrd 2868	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅))
71	45, 70	sselid 3937	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅)))
72	3, 4, 5, 6	ttukeylem3 10483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ suc (◡𝐹‘𝑎) ∈ On) → (𝐺‘suc (◡𝐹‘𝑎)) = if(suc (◡𝐹‘𝑎) = ∪ suc (◡𝐹‘𝑎), if(suc (◡𝐹‘𝑎) = ∅, 𝐵, ∪ (𝐺 “ suc (◡𝐹‘𝑎))), ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅))))
73	38, 72	syldan 602	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘suc (◡𝐹‘𝑎)) = if(suc (◡𝐹‘𝑎) = ∪ suc (◡𝐹‘𝑎), if(suc (◡𝐹‘𝑎) = ∅, 𝐵, ∪ (𝐺 “ suc (◡𝐹‘𝑎))), ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅))))
74		sucidg 6433	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)) → (◡𝐹‘𝑎) ∈ suc (◡𝐹‘𝑎))
75	34, 74	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ suc (◡𝐹‘𝑎))
76		ordirr 6368	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord (◡𝐹‘𝑎) → ¬ (◡𝐹‘𝑎) ∈ (◡𝐹‘𝑎))
77	36, 46, 76	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ (◡𝐹‘𝑎) ∈ (◡𝐹‘𝑎))
78		nelne1 3057	. . . . . . . . . . . . . . . . 17 ⊢ (((◡𝐹‘𝑎) ∈ suc (◡𝐹‘𝑎) ∧ ¬ (◡𝐹‘𝑎) ∈ (◡𝐹‘𝑎)) → suc (◡𝐹‘𝑎) ≠ (◡𝐹‘𝑎))
79	75, 77, 78	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ≠ (◡𝐹‘𝑎))
80	79, 48	neeqtrrd 3034	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ≠ ∪ suc (◡𝐹‘𝑎))
81	80	neneqd 2965	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ suc (◡𝐹‘𝑎) = ∪ suc (◡𝐹‘𝑎))
82	81	iffalsed 4494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → if(suc (◡𝐹‘𝑎) = ∪ suc (◡𝐹‘𝑎), if(suc (◡𝐹‘𝑎) = ∅, 𝐵, ∪ (𝐺 “ suc (◡𝐹‘𝑎))), ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅))) = ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅)))
83	73, 82	eqtrd 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘suc (◡𝐹‘𝑎)) = ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ if(((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴, {(𝐹‘∪ suc (◡𝐹‘𝑎))}, ∅)))
84	71, 83	eleqtrrd 2868	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (𝐺‘suc (◡𝐹‘𝑎)))
85	44, 84	sseldd 3940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
86	85	expr 461	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝑎 ∈ (𝑦 ∖ 𝐵) → 𝑎 ∈ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵)))))
87	86	ssrdv 3945	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝑦 ∖ 𝐵) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
88	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
89	87, 88	unssd 4147	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → ((𝑦 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
90	20, 89	sstrid 3950	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → 𝑦 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
91	17, 90	eqssd 3956	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) = 𝑦)
92	91	expr 461	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) = 𝑦))
93		npss 4070	. . . 4 ⊢ (¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦 ↔ ((𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) = 𝑦))
94	92, 93	sylibr 237	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)
95	94	ralrimiva 3157	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)
96		sseq2 3965	. . . 4 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵)))))
97		psseq1 4046	. . . . . 6 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (𝑥 ⊊ 𝑦 ↔ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
98	97	notbid 321	. . . . 5 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (¬ 𝑥 ⊊ 𝑦 ↔ ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
99	98	ralbidv 3188	. . . 4 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
100	96, 99	anbi12d 643	. . 3 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → ((𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) ↔ (𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∧ ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)))
101	100	rspcev 3584	. 2 ⊢ (((𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∈ 𝐴 ∧ (𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∧ ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))
102	8, 16, 95, 101	syl12anc 849	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101  ∀wal 1561   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907   ⊊ wpss 3908  ∅c0 4288  ifcif 4483  𝒫 cpw 4558  {csn 4585  ∪ cuni 4868   ↦ cmpt 5186  ^◡ccnv 5651  dom cdm 5652  ran crn 5653   “ cima 5655  Ord word 6349  Oncon0 6350  suc csuc 6352  ⟶wf 6521  –1-1-onto→wf1o 6524  ‘cfv 6525  recscrecs 8345  Fincfn 8931  cardccrd 9909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-1o 8441  df-en 8932  df-dom 8933  df-fin 8935  df-card 9913

This theorem is referenced by:  ttukey2g  10488