Theorem ttukeylem7 10413

Description: Lemma for ttukey 10416. (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 6841	. . . 4 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ V
2	1	sucid 6395	. . 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 10412	. . 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 10410	. . 3 ⊢ (𝜑 → (𝐺‘∅) = 𝐵)
10		0elon 6366	. . . . 5 ⊢ ∅ ∈ On
11		cardon 9844	. . . . 5 ⊢ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On
12		0ss 4349	. . . . 5 ⊢ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵))
13	10, 11, 12	3pm3.2i 1340	. . . 4 ⊢ (∅ ∈ On ∧ (card‘(∪ 𝐴 ∖ 𝐵)) ∈ On ∧ ∅ ⊆ (card‘(∪ 𝐴 ∖ 𝐵)))
14	3, 4, 5, 6	ttukeylem5 10411	. . . 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 3966	. 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 4127	. . . . . . . 8 ⊢ 𝑦 ⊆ (𝑦 ∪ 𝐵)
19		undif1 4425	. . . . . . . 8 ⊢ ((𝑦 ∖ 𝐵) ∪ 𝐵) = (𝑦 ∪ 𝐵)
20	18, 19	sseqtrri 3980	. . . . . . 7 ⊢ 𝑦 ⊆ ((𝑦 ∖ 𝐵) ∪ 𝐵)
21		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝜑)
22		f1ocnv 6780	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵) → ◡𝐹:(∪ 𝐴 ∖ 𝐵)–1-1-onto→(card‘(∪ 𝐴 ∖ 𝐵)))
23		f1of 6768	. . . . . . . . . . . . . . . . 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 4080	. . . . . . . . . . . . . . . . . 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 4863	. . . . . . . . . . . . . . . . 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 4081	. . . . . . . . . . . . . . . . 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 3909	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (∪ 𝐴 ∖ 𝐵))
34	25, 33	ffvelcdmd 7024	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (◡𝐹‘𝑎) ∈ (card‘(∪ 𝐴 ∖ 𝐵)))
35		onelon 6336	. . . . . . . . . . . . . 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 7749	. . . . . . . . . . . . 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 6424	. . . . . . . . . . . . 13 ⊢ Ord (card‘(∪ 𝐴 ∖ 𝐵))
41		ordsucss 7754	. . . . . . . . . . . . 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 10411	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘suc (◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
45		ssun2 4128	. . . . . . . . . . . . 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 6321	. . . . . . . . . . . . . . . . . 18 ⊢ ((◡𝐹‘𝑎) ∈ On → Ord (◡𝐹‘𝑎))
47		ordunisuc 7768	. . . . . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . 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 7217	. . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 = (𝐹‘∪ suc (◡𝐹‘𝑎)))
54		velsn 4591	. . . . . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) = (𝐺‘(◡𝐹‘𝑎)))
57		ordelss 6327	. . . . . . . . . . . . . . . . . . . . 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 10411	. . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘(◡𝐹‘𝑎)) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
61	56, 60	eqsstrd 3965	. . . . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐺‘∪ suc (◡𝐹‘𝑎)) ⊆ 𝑦)
64	53, 27	eqeltrrd 2834	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → (𝐹‘∪ suc (◡𝐹‘𝑎)) ∈ 𝑦)
65	64	snssd 4760	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → {(𝐹‘∪ suc (◡𝐹‘𝑎))} ⊆ 𝑦)
66	63, 65	unssd 4141	. . . . . . . . . . . . . . . 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 10408	. . . . . . . . . . . . . . . 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 4482	. . . . . . . . . . . . . 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 2836	. . . . . . . . . . . . 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 3928	. . . . . . . . . . . 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 10409	. . . . . . . . . . . . . 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 6394	. . . . . . . . . . . . . . . . . 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 6329	. . . . . . . . . . . . . . . . . 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 3026	. . . . . . . . . . . . . . . . 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 3003	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → suc (◡𝐹‘𝑎) ≠ ∪ suc (◡𝐹‘𝑎))
81	80	neneqd 2934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → ¬ suc (◡𝐹‘𝑎) = ∪ suc (◡𝐹‘𝑎))
82	81	iffalsed 4485	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . 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 2836	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦) ∧ 𝑎 ∈ (𝑦 ∖ 𝐵))) → 𝑎 ∈ (𝐺‘suc (◡𝐹‘𝑎)))
85	44, 84	sseldd 3931	. . . . . . . . . 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 3936	. . . . . . . 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 4141	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → ((𝑦 ∖ 𝐵) ∪ 𝐵) ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
90	20, 89	sstrid 3942	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊆ 𝑦)) → 𝑦 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))))
91	17, 90	eqssd 3948	. . . . 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 4062	. . . 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 3125	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) ⊊ 𝑦)
96		sseq2 3957	. . . 4 ⊢ (𝑥 = (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵))) → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ (𝐺‘(card‘(∪ 𝐴 ∖ 𝐵)))))
97		psseq1 4039	. . . . . 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 3156	. . . 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 3573	. 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 1539   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   ⊆ wss 3898   ⊊ wpss 3899  ∅c0 4282  ifcif 4474  𝒫 cpw 4549  {csn 4575  ∪ cuni 4858   ↦ cmpt 5174  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   “ cima 5622  Ord word 6310  Oncon0 6311  suc csuc 6313  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  recscrecs 8296  Fincfn 8875  cardccrd 9835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-1o 8391  df-en 8876  df-dom 8877  df-fin 8879  df-card 9839

This theorem is referenced by:  ttukey2g  10414