Theorem alexsubALTlem2 21852

Description: Lemma for alexsubALT 21855. 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 3597	. . . . . . . . . . . . 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 3753	. . . . . . . . . . . . . . 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 3627	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑎 ⊆ 𝑧 ↔ 𝑎 ⊆ 𝑤))
4		pweq 4161	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝑤 → 𝒫 𝑧 = 𝒫 𝑤)
5	4	ineq1d 3813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = 𝑤 → (𝒫 𝑧 ∩ Fin) = (𝒫 𝑤 ∩ Fin))
6	5	raleqdv 3144	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 ↔ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))
7	3, 6	anbi12d 747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑤 → ((𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ↔ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)))
8	7	elrab 3363	. . . . . . . . . . . . . . . 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 4193	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ {∅} ↔ 𝑤 = ∅)
10	8, 9	orbi12i 543	. . . . . . . . . . . . . . 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 264	. . . . . . . . . . . . . 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 4168	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ 𝒫 (fi‘𝑥) → 𝑤 ⊆ (fi‘𝑥))
13	12	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) → 𝑤 ⊆ (fi‘𝑥))
14		0ss 3972	. . . . . . . . . . . . . . . 16 ⊢ ∅ ⊆ (fi‘𝑥)
15		sseq1 3626	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = ∅ → (𝑤 ⊆ (fi‘𝑥) ↔ ∅ ⊆ (fi‘𝑥)))
16	14, 15	mpbiri 248	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = ∅ → 𝑤 ⊆ (fi‘𝑥))
17	13, 16	jaoi 394	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) ∨ 𝑤 = ∅) → 𝑤 ⊆ (fi‘𝑥))
18	11, 17	sylbi 207	. . . . . . . . . . . . 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 2965	. . . . . . . . . . 11 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → ∀𝑤 ∈ 𝑦 𝑤 ⊆ (fi‘𝑥))
21		unissb 4469	. . . . . . . . . . 11 ⊢ (∪ 𝑦 ⊆ (fi‘𝑥) ↔ ∀𝑤 ∈ 𝑦 𝑤 ⊆ (fi‘𝑥))
22	20, 21	sylibr 224	. . . . . . . . . 10 ⊢ (𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → ∪ 𝑦 ⊆ (fi‘𝑥))
23	22	adantr 481	. . . . . . . . 9 ⊢ ((𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∧ [⊊] Or 𝑦) → ∪ 𝑦 ⊆ (fi‘𝑥))
24	23	ad2antlr 763	. . . . . . . 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 6954	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
26	25	elpw 4164	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 (fi‘𝑥) ↔ ∪ 𝑦 ⊆ (fi‘𝑥))
27	24, 26	sylibr 224	. . . . . . 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 4463	. . . . . . . . . 10 ⊢ (∪ 𝑦 = ∅ ↔ 𝑦 ⊆ {∅})
29	28	notbii 310	. . . . . . . . 9 ⊢ (¬ ∪ 𝑦 = ∅ ↔ ¬ 𝑦 ⊆ {∅})
30		disjssun 4036	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 2808	. . . . . . . . . . 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 3931	. . . . . . . . . . . 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 3796	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . 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 264	. . . . . . . . . . . . . 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 804	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → 𝑎 ⊆ 𝑤)
38		simpl 473	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → 𝑤 ∈ 𝑦)
39		ssuni 4459	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ⊆ 𝑤 ∧ 𝑤 ∈ 𝑦) → 𝑎 ⊆ ∪ 𝑦)
40	37, 38, 39	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ 𝑦 ∧ (𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))) → 𝑎 ⊆ ∪ 𝑦)
41	36, 40	sylbi 207	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) → 𝑎 ⊆ ∪ 𝑦)
42	41	exlimiv 1858	. . . . . . . . . . . 12 ⊢ (∃𝑤 𝑤 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)}) → 𝑎 ⊆ ∪ 𝑦)
43	33, 42	sylbi 207	. . . . . . . . . . 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 764	. . . . . . . . 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	syl5bi 232	. . . . . . . 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 445	. . . . . . 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 8268	. . . . . . . . . 10 ⊢ (𝑛 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ↔ (𝑛 ⊆ ∪ 𝑦 ∧ 𝑛 ∈ Fin))
49		unieq 4444	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∪ ∅)
50		uni0 4465	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ ∅ = ∅
51	49, 50	syl6eq 2672	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∅)
52	51	necon3bi 2820	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ ∪ 𝑦 = ∅ → 𝑦 ≠ ∅)
53	52	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) → 𝑦 ≠ ∅)
54	53	ad2antrl 764	. . . . . . . . . . . . . . . 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)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → 𝑦 ≠ ∅)
55		simplrr 801	. . . . . . . . . . . . . . . 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)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → [⊊] Or 𝑦)
56		simprlr 803	. . . . . . . . . . . . . . . 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)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → 𝑛 ∈ Fin)
57		simprr 796	. . . . . . . . . . . . . . . 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)) ∧ ((¬ ∪ 𝑦 = ∅ ∧ 𝑛 ∈ Fin) ∧ 𝑛 ⊆ ∪ 𝑦)) → 𝑛 ⊆ ∪ 𝑦)
58		finsschain 8273	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ≠ ∅ ∧ [⊊] Or 𝑦) ∧ (𝑛 ∈ Fin ∧ 𝑛 ⊆ ∪ 𝑦)) → ∃𝑤 ∈ 𝑦 𝑛 ⊆ 𝑤)
59	54, 55, 56, 57, 58	syl22anc 1327	. . . . . . . . . . . . . . 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)) → ∃𝑤 ∈ 𝑦 𝑛 ⊆ 𝑤)
60	59	expr 643	. . . . . . . . . . . . . 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))
61		0elpw 4834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ 𝒫 𝑎
62		0fin 8188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∅ ∈ Fin
63		elin 3796	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∅ ∈ (𝒫 𝑎 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑎 ∧ ∅ ∈ Fin))
64	61, 62, 63	mpbir2an 955	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∅ ∈ (𝒫 𝑎 ∩ Fin)
65		unieq 4444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∪ ∅)
66	65	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = ∅ → (𝑋 = ∪ 𝑏 ↔ 𝑋 = ∪ ∅))
67	66	notbid 308	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = ∅ → (¬ 𝑋 = ∪ 𝑏 ↔ ¬ 𝑋 = ∪ ∅))
68	67	rspccv 3306	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (∅ ∈ (𝒫 𝑎 ∩ Fin) → ¬ 𝑋 = ∪ ∅))
69	64, 68	mpi 20	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑏 ∈ (𝒫 𝑎 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ¬ 𝑋 = ∪ ∅)
70		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑛 ∈ V
71	70	elpw 4164	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 ∈ 𝒫 𝑤 ↔ 𝑛 ⊆ 𝑤)
72		elin 3796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 ∈ (𝒫 𝑤 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑤 ∧ 𝑛 ∈ Fin))
73		unieq 4444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑏 = 𝑛 → ∪ 𝑏 = ∪ 𝑛)
74	73	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑏 = 𝑛 → (𝑋 = ∪ 𝑏 ↔ 𝑋 = ∪ 𝑛))
75	74	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑏 = 𝑛 → (¬ 𝑋 = ∪ 𝑏 ↔ ¬ 𝑋 = ∪ 𝑛))
76	75	rspccv 3306	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ∈ (𝒫 𝑤 ∩ Fin) → ¬ 𝑋 = ∪ 𝑛))
77	72, 76	syl5bir 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → ((𝑛 ∈ 𝒫 𝑤 ∧ 𝑛 ∈ Fin) → ¬ 𝑋 = ∪ 𝑛))
78	77	expd 452	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ∈ 𝒫 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛)))
79	71, 78	syl5bir 233	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ⊆ 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛)))
80	79	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛)))
81	80	ad2antll 765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛)))
82	81	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑋 = ∪ ∅ → ((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
83		sseq2 3627	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 = ∅ → (𝑛 ⊆ 𝑤 ↔ 𝑛 ⊆ ∅))
84		ss0 3974	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 ⊆ ∅ → 𝑛 = ∅)
85	83, 84	syl6bi 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = ∅ → (𝑛 ⊆ 𝑤 → 𝑛 = ∅))
86		unieq 4444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑛 = ∅ → ∪ 𝑛 = ∪ ∅)
87	86	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑛 = ∅ → (𝑋 = ∪ 𝑛 ↔ 𝑋 = ∪ ∅))
88	87	notbid 308	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = ∅ → (¬ 𝑋 = ∪ 𝑛 ↔ ¬ 𝑋 = ∪ ∅))
89	88	biimprcd 240	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑛 = ∅ → ¬ 𝑋 = ∪ 𝑛))
90	89	a1dd 50	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑛 = ∅ → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛)))
91	85, 90	syl9r 78	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑤 = ∅ → (𝑛 ⊆ 𝑤 → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛))))
92	91	com34 91	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑤 = ∅ → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
93	82, 92	jaod 395	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (¬ 𝑋 = ∪ ∅ → (((𝑤 ∈ 𝒫 (fi‘𝑥) ∧ (𝑎 ⊆ 𝑤 ∧ ∀𝑏 ∈ (𝒫 𝑤 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)) ∨ 𝑤 = ∅) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
94	11, 93	syl5bi 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑋 = ∪ ∅ → (𝑤 ∈ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
95	1, 94	sylan9r 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑋 = ∪ ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})) → (𝑤 ∈ 𝑦 → (𝑛 ∈ Fin → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
96	95	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑋 = ∪ ∅ ∧ 𝑦 ⊆ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅})) → (𝑛 ∈ Fin → (𝑤 ∈ 𝑦 → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛))))
97	69, 96	sylan 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀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))))
98	97	ad2ant2lr 784	. . . . . . . . . . . . . . . . 17 ⊢ ((((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))))
99	98	imp 445	. . . . . . . . . . . . . . . 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)))
100	99	adantrl 752	. . . . . . . . . . . . . . 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)) → (𝑤 ∈ 𝑦 → (𝑛 ⊆ 𝑤 → ¬ 𝑋 = ∪ 𝑛)))
101	100	rexlimdv 3030	. . . . . . . . . . . . . 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))
102	60, 101	syld 47	. . . . . . . . . . . . 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 → ¬ 𝑋 = ∪ 𝑛))
103	102	expr 643	. . . . . . . . . . . 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)))
104	103	com23 86	. . . . . . . . . . 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 ⊆ ∪ 𝑦 → (𝑛 ∈ Fin → ¬ 𝑋 = ∪ 𝑛)))
105	104	impd 447	. . . . . . . . . 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))
106	48, 105	syl5bi 232	. . . . . . . . 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))
107	106	ralrimiv 2965	. . . . . . . 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)
108		unieq 4444	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑏 → ∪ 𝑛 = ∪ 𝑏)
109	108	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑛 = 𝑏 → (𝑋 = ∪ 𝑛 ↔ 𝑋 = ∪ 𝑏))
110	109	notbid 308	. . . . . . . . 9 ⊢ (𝑛 = 𝑏 → (¬ 𝑋 = ∪ 𝑛 ↔ ¬ 𝑋 = ∪ 𝑏))
111	110	cbvralv 3171	. . . . . . . 8 ⊢ (∀𝑛 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑛 ↔ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)
112	107, 111	sylib 208	. . . . . . 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)
113	27, 47, 112	jca32 558	. . . . . 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)))
114	113	ex 450	. . . . 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))))
115		orcom 402	. . . . . 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 ∈ {∅}))
116	25	elsn 4192	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ {∅} ↔ ∪ 𝑦 = ∅)
117		sseq2 3627	. . . . . . . . . 10 ⊢ (𝑧 = ∪ 𝑦 → (𝑎 ⊆ 𝑧 ↔ 𝑎 ⊆ ∪ 𝑦))
118		pweq 4161	. . . . . . . . . . . 12 ⊢ (𝑧 = ∪ 𝑦 → 𝒫 𝑧 = 𝒫 ∪ 𝑦)
119	118	ineq1d 3813	. . . . . . . . . . 11 ⊢ (𝑧 = ∪ 𝑦 → (𝒫 𝑧 ∩ Fin) = (𝒫 ∪ 𝑦 ∩ Fin))
120	119	raleqdv 3144	. . . . . . . . . 10 ⊢ (𝑧 = ∪ 𝑦 → (∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏 ↔ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏))
121	117, 120	anbi12d 747	. . . . . . . . 9 ⊢ (𝑧 = ∪ 𝑦 → ((𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏) ↔ (𝑎 ⊆ ∪ 𝑦 ∧ ∀𝑏 ∈ (𝒫 ∪ 𝑦 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)))
122	121	elrab 3363	. . . . . . . 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)))
123	116, 122	orbi12i 543	. . . . . . 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))))
124		df-or 385	. . . . . . 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))))
125	123, 124	bitr2i 265	. . . . . 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)}))
126		elun 3753	. . . . . 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 ∈ {∅}))
127	115, 125, 126	3bitr4i 292	. . . . 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)} ∪ {∅}))
128	114, 127	sylib 208	. . . 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)} ∪ {∅}))
129	128	ex 450	. . 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)} ∪ {∅})))
130	129	alrimiv 1855	. 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)} ∪ {∅})))
131		fvex 6201	. . . . . 6 ⊢ (fi‘𝑥) ∈ V
132	131	pwex 4848	. . . . 5 ⊢ 𝒫 (fi‘𝑥) ∈ V
133	132	rabex 4813	. . . 4 ⊢ {𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∈ V
134		p0ex 4853	. . . 4 ⊢ {∅} ∈ V
135	133, 134	unex 6956	. . 3 ⊢ ({𝑧 ∈ 𝒫 (fi‘𝑥) ∣ (𝑎 ⊆ 𝑧 ∧ ∀𝑏 ∈ (𝒫 𝑧 ∩ Fin) ¬ 𝑋 = ∪ 𝑏)} ∪ {∅}) ∈ V
136	135	zorn 9329	. 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)
137	130, 136	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)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574   ⊊ wpss 3575  ∅c0 3915  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436   Or wor 5034  ‘cfv 5888   [⊊] crpss 6936  Fincfn 7955  ficfi 8316  topGenctg 16098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-rpss 6937  df-om 7066  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-en 7956  df-fin 7959  df-card 8765  df-ac 8939

This theorem is referenced by:  alexsubALTlem4  21854