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Theorem alexsubALTlem2 23939

Description: Lemma for alexsubALT 23942. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)

**Hypothesis**
Ref	Expression
alexsubALT.1	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
alexsubALTlem2	⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ¬ 𝑢 ⊊ 𝑣)

Distinct variable groups: 𝑎,𝑏,𝑐,𝑑,𝑢,𝑣,𝑥,𝑧,𝐽 𝑋,𝑎,𝑏,𝑐,𝑑,𝑢,𝑣,𝑥,𝑧

Proof of Theorem alexsubALTlem2 Dummy variables 𝑛 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3971	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → (𝑤 ∈ 𝑦 → 𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})))
2		elun 4144	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ↔ (𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∨ 𝑤 ∈ {∅}))
3		sseq2 4004	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑎 ⊆ 𝑧 ↔ 𝑎 ⊆ 𝑤))
4		pweq 4612	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
5	4	ineq1d 4207	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑤 ∩ Fin))
6	5	raleqdv 3320	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 ↔ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))
7	3, 6	anbi12d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → ((𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ↔ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)))
8	7	elrab 3680	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ↔ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)))
9		velsn 4640	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {∅} ↔ 𝑤 = ∅)
10	8, 9	orbi12i 913	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∨ 𝑤 ∈ {∅}) ↔ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) ∨ 𝑤 = ∅))
11	2, 10	bitri 275	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ↔ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) ∨ 𝑤 = ∅))
12		elpwi 4605	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝒫 (fi‘𝑥) → 𝑤 ⊆ (fi‘𝑥))
13	12	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) → 𝑤 ⊆ (fi‘𝑥))
14		0ss 4392	. . . . . . . . . . . . . . . 16 ⊢ ∅ ⊆ (fi‘𝑥)
15		sseq1 4003	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ∅ → (𝑤 ⊆ (fi‘𝑥) ↔ ∅ ⊆ (fi‘𝑥)))
16	14, 15	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∅ → 𝑤 ⊆ (fi‘𝑥))
17	13, 16	jaoi 856	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) ∨ 𝑤 = ∅) → 𝑤 ⊆ (fi‘𝑥))
18	11, 17	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → 𝑤 ⊆ (fi‘𝑥))
19	1, 18	syl6 35	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → (𝑤 ∈ 𝑦 → 𝑤 ⊆ (fi‘𝑥)))
20	19	ralrimiv 3140	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → ∀𝑤 ∈ 𝑦 𝑤 ⊆ (fi‘𝑥))
21		unissb 4937	. . . . . . . . . . 11 ⊢ (∪ 𝑦 ⊆ (fi‘𝑥) ↔ ∀𝑤 ∈ 𝑦 𝑤 ⊆ (fi‘𝑥))
22	20, 21	sylibr 233	. . . . . . . . . 10 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → ∪ 𝑦 ⊆ (fi‘𝑥))
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦) → ∪ 𝑦 ⊆ (fi‘𝑥))
24	23	ad2antlr 726	. . . . . . . 8 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → ∪ 𝑦 ⊆ (fi‘𝑥))
25		vuniex 7738	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
26	25	elpw 4602	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ↔ ∪ 𝑦 ⊆ (fi‘𝑥))
27	24, 26	sylibr 233	. . . . . . 7 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → ∪ 𝑦 ∈ 𝒫 (fi‘𝑥))
28		uni0b 4931	. . . . . . . . . 10 ⊢ (∪ 𝑦 = ∅ ↔ 𝑦 ⊆ {∅})
29	28	notbii 320	. . . . . . . . 9 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ¬ 𝑦 ⊆ {∅})
30		disjssun 4463	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) = ∅ → (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ↔ 𝑦 ⊆ {∅}))
31	30	biimpcd 248	. . . . . . . . . . . 12 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) = ∅ → 𝑦 ⊆ {∅}))
32	31	necon3bd 2949	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → (¬ 𝑦 ⊆ {∅} → (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ≠ ∅))
33		n0 4342	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}))
34		elin 3960	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ↔ (𝑤 ∈ 𝑦 ∧ 𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}))
35	8	anbi2i 622	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝑦 ∧ 𝑤 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ↔ (𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))))
36	34, 35	bitri 275	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ↔ (𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))))
37		simprrl 780	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → 𝑎 ⊆ 𝑤)
38		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → 𝑤 ∈ 𝑦)
39		ssuni 4930	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ 𝑤 ∧ 𝑤 ∈ 𝑦) → 𝑎 ⊆ ∪ 𝑦)
40	37, 38, 39	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → 𝑎 ⊆ ∪ 𝑦)
41	36, 40	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) → 𝑎 ⊆ ∪ 𝑦)
42	41	exlimiv 1926	. . . . . . . . . . . 12 ⊢ (∃𝑤 𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) → 𝑎 ⊆ ∪ 𝑦)
43	33, 42	sylbi 216	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ≠ ∅ → 𝑎 ⊆ ∪ 𝑦)
44	32, 43	syl6 35	. . . . . . . . . 10 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → (¬ 𝑦 ⊆ {∅} → 𝑎 ⊆ ∪ 𝑦))
45	44	ad2antrl 727	. . . . . . . . 9 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) → (¬ 𝑦 ⊆ {∅} → 𝑎 ⊆ ∪ 𝑦))
46	29, 45	biimtrid 241	. . . . . . . 8 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) → (¬ ∪ 𝑦 = ∅ → 𝑎 ⊆ ∪ 𝑦))
47	46	imp 406	. . . . . . 7 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → 𝑎 ⊆ ∪ 𝑦)
48		elfpw 9370	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ↔ (𝑛 ⊆ ∪ 𝑦 ∧ 𝑛 ∈ Fin))
49		unieq 4914	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∪ ∅)
50		uni0 4933	. . . . . . . . . . . . . . . . . . 19 ⊢ ∪ ∅ = ∅
51	49, 50	eqtrdi 2783	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∅)
52	51	necon3bi 2962	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∪ 𝑦 = ∅ → 𝑦 ≠ ∅)
53	52	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) → 𝑦 ≠ ∅)
54	53	ad2antrl 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → 𝑦 ≠ ∅)
55		simplrr 777	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → [⊊] Or 𝑦)
56		simprlr 779	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → 𝑛 ∈ Fin)
57		simprr 772	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → 𝑛 ⊆ ∪ 𝑦)
58		finsschain 9375	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ∧ (𝑛 ∈ Fin ∧ 𝑛 ⊆ ∪ 𝑦)) → ∃𝑤 ∈ 𝑦 𝑛 ⊆ 𝑤)
59	54, 55, 56, 57, 58	syl22anc 838	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → ∃𝑤 ∈ 𝑦 𝑛 ⊆ 𝑤)
60	59	expr 456	. . . . . . . . . . . . 13 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ (¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (𝑛 ⊆ ∪ 𝑦 → ∃𝑤 ∈ 𝑦 𝑛 ⊆ 𝑤))
61		0elpw 5350	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ 𝒫 𝑎
62		0fin 9187	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ Fin
63	61, 62	elini 4189	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ (𝒫 𝑎 ∩ Fin)
64		unieq 4914	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∪ ∅)
65	64	eqeq2d 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = ∅ → (𝑋 = ∪ 𝑏 ↔ 𝑋 = ∪ ∅))
66	65	notbid 318	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 = ∅ → (¬ 𝑋 = ∪ 𝑏 ↔ ¬ 𝑋 = ∪ ∅))
67	66	rspccv 3604	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (∅ ∈ (𝒫 𝑎 ∩ Fin) → ¬ 𝑋 = ∪ ∅))
68	63, 67	mpi 20	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ¬ 𝑋 = ∪ ∅)
69		velpw 4603	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ∈ 𝒫 𝑤 ↔ 𝑛 ⊆ 𝑤)
70		elin 3960	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 ∈ (𝒫 𝑤 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑤 ∧ 𝑛 ∈ Fin))
71		unieq 4914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 = 𝑛 → ∪ 𝑏 = ∪ 𝑛)
72	71	eqeq2d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑛 → (𝑋 = ∪ 𝑏 ↔ 𝑋 = ∪ 𝑛))
73	72	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑏 = 𝑛 → (¬ 𝑋 = ∪ 𝑏 ↔ ¬ 𝑋 = ∪ 𝑛))
74	73	rspccv 3604	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ∈ (𝒫 𝑤 ∩ Fin) → ¬ 𝑋 = ∪ 𝑛))
75	70, 74	biimtrrid 242	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ((𝑛 ∈ 𝒫 𝑤 ∧ 𝑛 ∈ Fin) → ¬ 𝑋 = ∪ 𝑛))
76	75	expd 415	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ∈ 𝒫 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛)))
77	69, 76	biimtrrid 242	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ⊆ 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛)))
78	77	com23 86	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛)))
79	78	ad2antll 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛)))
80	79	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑋 = ∪ ∅ → ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
81		sseq2 4004	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = ∅ → (𝑛 ⊆ 𝑤 ↔ 𝑛 ⊆ ∅))
82		ss0 4394	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ⊆ ∅ → 𝑛 = ∅)
83	81, 82	biimtrdi 252	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = ∅ → (𝑛 ⊆ 𝑤 → 𝑛 = ∅))
84		unieq 4914	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = ∅ → ∪ 𝑛 = ∪ ∅)
85	84	eqeq2d 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = ∅ → (𝑋 = ∪ 𝑛 ↔ 𝑋 = ∪ ∅))
86	85	notbid 318	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = ∅ → (¬ 𝑋 = ∪ 𝑛 ↔ ¬ 𝑋 = ∪ ∅))
87	86	biimprcd 249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑛 = ∅ → ¬ 𝑋 = ∪ 𝑛))
88	87	a1dd 50	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑛 = ∅ → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛)))
89	83, 88	syl9r 78	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑤 = ∅ → (𝑛 ⊆ 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛))))
90	89	com34 91	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑤 = ∅ → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
91	80, 90	jaod 858	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑋 = ∪ ∅ → (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) ∨ 𝑤 = ∅) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
92	11, 91	biimtrid 241	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
93	1, 92	sylan9r 508	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑋 = ∪ ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})) → (𝑤 ∈ 𝑦 → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
94	93	com23 86	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑋 = ∪ ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})) → (𝑛 ∈ Fin → (𝑤 ∈ 𝑦 → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
95	68, 94	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})) → (𝑛 ∈ Fin → (𝑤 ∈ 𝑦 → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
96	95	ad2ant2lr 747	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) → (𝑛 ∈ Fin → (𝑤 ∈ 𝑦 → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
97	96	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ 𝑛 ∈ Fin) → (𝑤 ∈ 𝑦 → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛)))
98	97	adantrl 715	. . . . . . . . . . . . . 14 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ (¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (𝑤 ∈ 𝑦 → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛)))
99	98	rexlimdv 3148	. . . . . . . . . . . . 13 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ (¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (∃𝑤 ∈ 𝑦 𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))
100	60, 99	syld 47	. . . . . . . . . . . 12 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ (¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin)) → (𝑛 ⊆ ∪ 𝑦 → ¬ 𝑋 = ∪ 𝑛))
101	100	expr 456	. . . . . . . . . . 11 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → (𝑛 ∈ Fin → (𝑛 ⊆ ∪ 𝑦 → ¬ 𝑋 = ∪ 𝑛)))
102	101	impcomd 411	. . . . . . . . . 10 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → ((𝑛 ⊆ ∪ 𝑦 ∧ 𝑛 ∈ Fin) → ¬ 𝑋 = ∪ 𝑛))
103	48, 102	biimtrid 241	. . . . . . . . 9 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → (𝑛 ∈ (𝒫 ∪ 𝑦 ∩ Fin) → ¬ 𝑋 = ∪ 𝑛))
104	103	ralrimiv 3140	. . . . . . . 8 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → ∀𝑛 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑛)
105		unieq 4914	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑏 → ∪ 𝑛 = ∪ 𝑏)
106	105	eqeq2d 2738	. . . . . . . . . 10 ⊢ (𝑛 = 𝑏 → (𝑋 = ∪ 𝑛 ↔ 𝑋 = ∪ 𝑏))
107	106	notbid 318	. . . . . . . . 9 ⊢ (𝑛 = 𝑏 → (¬ 𝑋 = ∪ 𝑛 ↔ ¬ 𝑋 = ∪ 𝑏))
108	107	cbvralvw 3229	. . . . . . . 8 ⊢ (∀𝑛 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑛 ↔ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)
109	104, 108	sylib 217	. . . . . . 7 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)
110	27, 47, 109	jca32 515	. . . . . 6 ⊢ (((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) ∧ ¬ ∪ 𝑦 = ∅) → (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)))
111	110	ex 412	. . . . 5 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) → (¬ ∪ 𝑦 = ∅ → (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))))
112		orcom 869	. . . . . 6 ⊢ ((∪ 𝑦 ∈ {∅} ∨ ∪ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ↔ (∪ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∨ ∪ 𝑦 ∈ {∅}))
113	25	elsn 4639	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ {∅} ↔ ∪ 𝑦 = ∅)
114		sseq2 4004	. . . . . . . . . 10 ⊢ (𝑧 = ∪ 𝑦 → (𝑎 ⊆ 𝑧 ↔ 𝑎 ⊆ ∪ 𝑦))
115		pweq 4612	. . . . . . . . . . . 12 ⊢ (𝑧 = ∪ 𝑦 → 𝒫 𝑧 = 𝒫 ∪ 𝑦)
116	115	ineq1d 4207	. . . . . . . . . . 11 ⊢ (𝑧 = ∪ 𝑦 → (𝒫 𝑧 ∩ Fin) = (𝒫 ∪ 𝑦 ∩ Fin))
117	116	raleqdv 3320	. . . . . . . . . 10 ⊢ (𝑧 = ∪ 𝑦 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 ↔ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))
118	114, 117	anbi12d 630	. . . . . . . . 9 ⊢ (𝑧 = ∪ 𝑦 → ((𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ↔ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)))
119	118	elrab 3680	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ↔ (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)))
120	113, 119	orbi12i 913	. . . . . . 7 ⊢ ((∪ 𝑦 ∈ {∅} ∨ ∪ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) ↔ (∪ 𝑦 = ∅ ∨ (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))))
121		df-or 847	. . . . . . 7 ⊢ ((∪ 𝑦 = ∅ ∨ (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) ↔ (¬ ∪ 𝑦 = ∅ → (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))))
122	120, 121	bitr2i 276	. . . . . 6 ⊢ ((¬ ∪ 𝑦 = ∅ → (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) ↔ (∪ 𝑦 ∈ {∅} ∨ ∪ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}))
123		elun 4144	. . . . . 6 ⊢ (∪ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ↔ (∪ 𝑦 ∈ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∨ ∪ 𝑦 ∈ {∅}))
124	112, 122, 123	3bitr4i 303	. . . . 5 ⊢ ((¬ ∪ 𝑦 = ∅ → (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) ↔ ∪ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}))
125	111, 124	sylib 217	. . . 4 ⊢ ((((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ∧ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦)) → ∪ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}))
126	125	ex 412	. . 3 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) → ((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})))
127	126	alrimiv 1923	. 2 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) → ∀𝑦((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})))
128		fvex 6904	. . . . . 6 ⊢ (fi‘𝑥) ∈ V
129	128	pwex 5374	. . . . 5 ⊢ 𝒫 (fi‘𝑥) ∈ V
130	129	rabex 5328	. . . 4 ⊢ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∈ V
131		p0ex 5378	. . . 4 ⊢ {∅} ∈ V
132	130, 131	unex 7742	. . 3 ⊢ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∈ V
133	132	zorn 10522	. 2 ⊢ (∀𝑦((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦) → ∪ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ¬ 𝑢 ⊊ 𝑣)
134	127, 133	syl 17	1 ⊢ (((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ∧ 𝑎 ∈ 𝒫 (fi‘𝑥)) ∧ ∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) → ∃𝑢 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})∀𝑣 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ¬ 𝑢 ⊊ 𝑣)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 ∀wal 1532 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 {crab 3427 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ⊊ wpss 3945 ∅c0 4318 𝒫 cpw 4598 {csn 4624 ∪ cuni 4903 Or wor 5583 ‘cfv 6542 [⊊] crpss 7721 Fincfn 8955 ficfi 9425 topGenctg 17410

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-ac2 10478

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-rpss 7722 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-en 8956 df-fin 8959 df-card 9954 df-ac 10131

This theorem is referenced by: alexsubALTlem4 23941

W3C validator