Theorem ttukeylem7 10202

Description: Lemma for ttukey 10205. (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 6769	. . . 4 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
2	1	sucid 6330	. . 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 10201	. . 3 ⊢ ((𝜑 ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ suc (card‘(∪ 𝐴 ∖ 𝐵))) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∈ 𝐴)
8	2, 7	mpan2 687	. 2 ⊢ (𝜑 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∈ 𝐴)
9	3, 4, 5, 6	ttukeylem4 10199	. . 3 ⊢ (𝜑 → (𝐺‘∅) = 𝐵)
10		0elon 6304	. . . . 5 ⊢ ∅ ∈ On
11		cardon 9633	. . . . 5 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On
12		0ss 4327	. . . . 5 ⊢ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵))
13	10, 11, 12	3pm3.2i 1337	. . . 4 ⊢ (∅ ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
14	3, 4, 5, 6	ttukeylem5 10200	. . . 4 ⊢ ((𝜑 ∧ (∅ ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))) → (𝐺‘∅) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
15	13, 14	mpan2 687	. . 3 ⊢ (𝜑 → (𝐺‘∅) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
16	9, 15	eqsstrrd 3956	. 2 ⊢ (𝜑 → 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
17		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)
18		ssun1 4102	. . . . . . . 8 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝐵)
19		undif1 4406	. . . . . . . 8 ⊢ ((𝑦 ∖ 𝐵) ∪ 𝐵) = (𝑦 ∪ 𝐵)
20	18, 19	sseqtrri 3954	. . . . . . 7 ⊢ 𝑦 ⊆ ((𝑦 ∖ 𝐵) ∪ 𝐵)
21		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝜑)
22		f1ocnv 6712	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)–1-1-onto→(card‘(∪ 𝐴 ∖ 𝐵)))
23		f1of 6700	. . . . . . . . . . . . . . . . 17 ⊢ (◡𝐹:(∪ 𝐴 ∖ 𝐵)–1-1-onto→(card‘(∪ 𝐴 ∖ 𝐵)) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)⟶(card‘(∪ 𝐴 ∖ 𝐵)))
24	3, 22, 23	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ◡𝐹:(∪ 𝐴 ∖ 𝐵)⟶(card‘(∪ 𝐴 ∖ 𝐵)))
25	24	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)⟶(card‘(∪ 𝐴 ∖ 𝐵)))
26		eldifi 4057	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑦 ∖ 𝐵) → 𝑎 ∈ 𝑦)
27	26	ad2antll 725	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ 𝑦)
28		simprll 775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑦 ∈ 𝐴)
29		elunii 4841	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑎 ∈ ∪ 𝐴)
30	27, 28, 29	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ ∪ 𝐴)
31		eldifn 4058	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝑦 ∖ 𝐵) → ¬ 𝑎 ∈ 𝐵)
32	31	ad2antll 725	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ 𝑎 ∈ 𝐵)
33	30, 32	eldifd 3894	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (∪ 𝐴 ∖ 𝐵))
34	25, 33	ffvelrnd 6944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)))
35		onelon 6276	. . . . . . . . . . . . . 14 ⊢ (((card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ On)
36	11, 34, 35	sylancr 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ On)
37		suceloni 7635	. . . . . . . . . . . . 13 ⊢ ((◡𝐹‘𝑎) ∈ On → suc (◡𝐹‘𝑎) ∈ On)
38	36, 37	syl 17	. . . . . . . . . . . 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 6356	. . . . . . . . . . . . 13 ⊢ Ord (card‘(∪ 𝐴 ∖ 𝐵))
41		ordsucss 7640	. . . . . . . . . . . . 13 ⊢ (Ord (card‘(∪ 𝐴 ∖ 𝐵)) → ((◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)) → suc (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵))))
42	40, 34, 41	mpsyl 68	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
43	3, 4, 5, 6	ttukeylem5 10200	. . . . . . . . . . . 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 1370	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
45		ssun2 4103	. . . . . . . . . . . . 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 6261	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝐹‘𝑎) ∈ On → Ord (◡𝐹‘𝑎))
47		ordunisuc 7654	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord (◡𝐹‘𝑎) → ∪ suc (◡𝐹‘𝑎) = (◡𝐹‘𝑎))
48	36, 46, 47	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ∪ suc (◡𝐹‘𝑎) = (◡𝐹‘𝑎))
49	48	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘∪ suc (◡𝐹‘𝑎)) = (𝐹‘(◡𝐹‘𝑎)))
50	3	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵))
51		f1ocnvfv2 7130	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝑎 ∈ (∪ 𝐴 ∖ 𝐵)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
52	50, 33, 51	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
53	49, 52	eqtr2d 2779	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 = (𝐹‘∪ suc (◡𝐹‘𝑎)))
54		velsn 4574	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {(𝐹‘∪ suc (◡𝐹‘𝑎))} ↔ 𝑎 = (𝐹‘∪ suc (◡𝐹‘𝑎)))
55	53, 54	sylibr 233	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ {(𝐹‘∪ suc (◡𝐹‘𝑎))})
56	48	fveq2d 6760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) = (𝐺‘(◡𝐹‘𝑎)))
57		ordelss 6267	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Ord (card‘(∪ 𝐴 ∖ 𝐵)) ∧ (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵))) → (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
58	40, 34, 57	sylancr 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
59	3, 4, 5, 6	ttukeylem5 10200	. . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘(◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
61	56, 60	eqsstrd 3955	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
62		simprlr 776	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)
63	61, 62	sstrd 3927	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) ⊆ 𝑦)
64	53, 27	eqeltrrd 2840	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘∪ suc (◡𝐹‘𝑎)) ∈ 𝑦)
65	64	snssd 4739	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → {(𝐹‘∪ suc (◡𝐹‘𝑎))} ⊆ 𝑦)
66	63, 65	unssd 4116	. . . . . . . . . . . . . . . 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 10197	. . . . . . . . . . . . . . . 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 833	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴)
69	68	iftrued 4464	. . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . 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 3915	. . . . . . . . . . . 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 10198	. . . . . . . . . . . . . 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 590	. . . . . . . . . . . . 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 6329	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)) → (◡𝐹‘𝑎) ∈ suc (◡𝐹‘𝑎))
75	34, 74	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ suc (◡𝐹‘𝑎))
76		ordirr 6269	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord (◡𝐹‘𝑎) → ¬ (◡𝐹‘𝑎) ∈ (◡𝐹‘𝑎))
77	36, 46, 76	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ (◡𝐹‘𝑎) ∈ (◡𝐹‘𝑎))
78		nelne1 3040	. . . . . . . . . . . . . . . . 17 ⊢ (((◡𝐹‘𝑎) ∈ suc (◡𝐹‘𝑎) ∧ ¬ (◡𝐹‘𝑎) ∈ (◡𝐹‘𝑎)) → suc (◡𝐹‘𝑎) ≠ (◡𝐹‘𝑎))
79	75, 77, 78	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ≠ (◡𝐹‘𝑎))
80	79, 48	neeqtrrd 3017	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ≠ ∪ suc (◡𝐹‘𝑎))
81	80	neneqd 2947	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ suc (◡𝐹‘𝑎) = ∪ suc (◡𝐹‘𝑎))
82	81	iffalsed 4467	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 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 2842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (𝐺‘suc (◡𝐹‘𝑎)))
85	44, 84	sseldd 3918	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
86	85	expr 456	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝑎 ∈ (𝑦 ∖ 𝐵) → 𝑎 ∈ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵)))))
87	86	ssrdv 3923	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝑦 ∖ 𝐵) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
88	16	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
89	87, 88	unssd 4116	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → ((𝑦 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
90	20, 89	sstrid 3928	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → 𝑦 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
91	17, 90	eqssd 3934	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) = 𝑦)
92	91	expr 456	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) = 𝑦))
93		npss 4041	. . . 4 ⊢ (¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦 ↔ ((𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) = 𝑦))
94	92, 93	sylibr 233	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)
95	94	ralrimiva 3107	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)
96		sseq2 3943	. . . 4 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵)))))
97		psseq1 4018	. . . . . 6 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (𝑥 ⊊ 𝑦 ↔ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
98	97	notbid 317	. . . . 5 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (¬ 𝑥 ⊊ 𝑦 ↔ ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
99	98	ralbidv 3120	. . . 4 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
100	96, 99	anbi12d 630	. . 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 3552	. 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 833	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883   ⊊ wpss 3884  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   ↦ cmpt 5153  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583  Ord word 6250  Oncon0 6251  suc csuc 6253  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  recscrecs 8172  Fincfn 8691  cardccrd 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-fin 8695  df-card 9628

This theorem is referenced by:  ttukey2g  10203