Theorem ttukeylem7 10556

Description: Lemma for ttukey 10559. (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 6918	. . . 4 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
2	1	sucid 6465	. . 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 10555	. . 3 ⊢ ((𝜑 ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ suc (card‘(∪ 𝐴 ∖ 𝐵))) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∈ 𝐴)
8	2, 7	mpan2 691	. 2 ⊢ (𝜑 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ∈ 𝐴)
9	3, 4, 5, 6	ttukeylem4 10553	. . 3 ⊢ (𝜑 → (𝐺‘∅) = 𝐵)
10		0elon 6437	. . . . 5 ⊢ ∅ ∈ On
11		cardon 9985	. . . . 5 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On
12		0ss 4399	. . . . 5 ⊢ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵))
13	10, 11, 12	3pm3.2i 1339	. . . 4 ⊢ (∅ ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
14	3, 4, 5, 6	ttukeylem5 10554	. . . 4 ⊢ ((𝜑 ∧ (∅ ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))) → (𝐺‘∅) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
15	13, 14	mpan2 691	. . 3 ⊢ (𝜑 → (𝐺‘∅) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
16	9, 15	eqsstrrd 4018	. 2 ⊢ (𝜑 → 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
17		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)
18		ssun1 4177	. . . . . . . 8 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝐵)
19		undif1 4475	. . . . . . . 8 ⊢ ((𝑦 ∖ 𝐵) ∪ 𝐵) = (𝑦 ∪ 𝐵)
20	18, 19	sseqtrri 4032	. . . . . . 7 ⊢ 𝑦 ⊆ ((𝑦 ∖ 𝐵) ∪ 𝐵)
21		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝜑)
22		f1ocnv 6859	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)–1-1-onto→(card‘(∪ 𝐴 ∖ 𝐵)))
23		f1of 6847	. . . . . . . . . . . . . . . . 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 4130	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (𝑦 ∖ 𝐵) → 𝑎 ∈ 𝑦)
27	26	ad2antll 729	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ 𝑦)
28		simprll 778	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑦 ∈ 𝐴)
29		elunii 4911	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑎 ∈ ∪ 𝐴)
30	27, 28, 29	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ ∪ 𝐴)
31		eldifn 4131	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (𝑦 ∖ 𝐵) → ¬ 𝑎 ∈ 𝐵)
32	31	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ 𝑎 ∈ 𝐵)
33	30, 32	eldifd 3961	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (∪ 𝐴 ∖ 𝐵))
34	25, 33	ffvelcdmd 7104	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)))
35		onelon 6408	. . . . . . . . . . . . . 14 ⊢ (((card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ On)
36	11, 34, 35	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ On)
37		onsuc 7832	. . . . . . . . . . . . 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 6494	. . . . . . . . . . . . 13 ⊢ Ord (card‘(∪ 𝐴 ∖ 𝐵))
41		ordsucss 7839	. . . . . . . . . . . . 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 10554	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
45		ssun2 4178	. . . . . . . . . . . . 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 6393	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝐹‘𝑎) ∈ On → Ord (◡𝐹‘𝑎))
47		ordunisuc 7853	. . . . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . 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 7298	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) ∧ 𝑎 ∈ (∪ 𝐴 ∖ 𝐵)) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
52	50, 33, 51	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
53	49, 52	eqtr2d 2777	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 = (𝐹‘∪ suc (◡𝐹‘𝑎)))
54		velsn 4641	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ {(𝐹‘∪ suc (◡𝐹‘𝑎))} ↔ 𝑎 = (𝐹‘∪ suc (◡𝐹‘𝑎)))
55	53, 54	sylibr 234	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ {(𝐹‘∪ suc (◡𝐹‘𝑎))})
56	48	fveq2d 6909	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) = (𝐺‘(◡𝐹‘𝑎)))
57		ordelss 6399	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Ord (card‘(∪ 𝐴 ∖ 𝐵)) ∧ (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵))) → (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
58	40, 34, 57	sylancr 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
59	3, 4, 5, 6	ttukeylem5 10554	. . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘(◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
61	56, 60	eqsstrd 4017	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
62		simprlr 779	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)
63	61, 62	sstrd 3993	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) ⊆ 𝑦)
64	53, 27	eqeltrrd 2841	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘∪ suc (◡𝐹‘𝑎)) ∈ 𝑦)
65	64	snssd 4808	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → {(𝐹‘∪ suc (◡𝐹‘𝑎))} ⊆ 𝑦)
66	63, 65	unssd 4191	. . . . . . . . . . . . . . . 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 10551	. . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ((𝐺‘∪ suc (◡𝐹‘𝑎)) ∪ {(𝐹‘∪ suc (◡𝐹‘𝑎))}) ∈ 𝐴)
69	68	iftrued 4532	. . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . 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 3980	. . . . . . . . . . . 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 10552	. . . . . . . . . . . . . 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 591	. . . . . . . . . . . . 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 6464	. . . . . . . . . . . . . . . . . 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 6401	. . . . . . . . . . . . . . . . . 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 3038	. . . . . . . . . . . . . . . . 17 ⊢ (((◡𝐹‘𝑎) ∈ suc (◡𝐹‘𝑎) ∧ ¬ (◡𝐹‘𝑎) ∈ (◡𝐹‘𝑎)) → suc (◡𝐹‘𝑎) ≠ (◡𝐹‘𝑎))
79	75, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ≠ (◡𝐹‘𝑎))
80	79, 48	neeqtrrd 3014	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ≠ ∪ suc (◡𝐹‘𝑎))
81	80	neneqd 2944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ suc (◡𝐹‘𝑎) = ∪ suc (◡𝐹‘𝑎))
82	81	iffalsed 4535	. . . . . . . . . . . . 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 2776	. . . . . . . . . . . 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 2843	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (𝐺‘suc (◡𝐹‘𝑎)))
85	44, 84	sseldd 3983	. . . . . . . . . 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 3988	. . . . . . . 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 4191	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → ((𝑦 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
90	20, 89	sstrid 3994	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → 𝑦 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
91	17, 90	eqssd 4000	. . . . 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 4112	. . . 4 ⊢ (¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦 ↔ ((𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦 → (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) = 𝑦))
94	92, 93	sylibr 234	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)
95	94	ralrimiva 3145	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)
96		sseq2 4009	. . . 4 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵)))))
97		psseq1 4089	. . . . . 6 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (𝑥 ⊊ 𝑦 ↔ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
98	97	notbid 318	. . . . 5 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (¬ 𝑥 ⊊ 𝑦 ↔ ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
99	98	ralbidv 3177	. . . 4 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦))
100	96, 99	anbi12d 632	. . 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 3621	. 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 836	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∖ cdif 3947   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950   ⊊ wpss 3951  ∅c0 4332  ifcif 4524  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906   ↦ cmpt 5224  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   “ cima 5687  Ord word 6382  Oncon0 6383  suc csuc 6385  ⟶wf 6556  –1-1-onto→wf1o 6559  ‘cfv 6560  recscrecs 8411  Fincfn 8986  cardccrd 9976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-en 8987  df-dom 8988  df-fin 8990  df-card 9980

This theorem is referenced by:  ttukey2g  10557