Theorem onsucsuccmpi 37201

Description: The successor of a successor ordinal number is a compact topology, proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 18-Oct-2015.)

Hypothesis

Ref	Expression
onsucsuccmpi.1	⊢ 𝐴 ∈ On

Assertion

Ref	Expression
onsucsuccmpi	⊢ suc suc 𝐴 ∈ Comp

Proof of Theorem onsucsuccmpi

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsucsuccmpi.1	. . . 4 ⊢ 𝐴 ∈ On
2	1	onsuci 7839	. . 3 ⊢ suc 𝐴 ∈ On
3		onsuctop 37191	. . 3 ⊢ (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
4	2, 3	ax-mp 5	. 2 ⊢ suc suc 𝐴 ∈ Top
5	1	onirri 6470	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
6	1, 1	onsucssi 7841	. . . . . . 7 ⊢ (𝐴 ∈ 𝐴 ↔ suc 𝐴 ⊆ 𝐴)
7	5, 6	mtbi 325	. . . . . 6 ⊢ ¬ suc 𝐴 ⊆ 𝐴
8		sseq1 3956	. . . . . 6 ⊢ (suc 𝐴 = ∪ 𝑦 → (suc 𝐴 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴))
9	7, 8	mtbii 329	. . . . 5 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ ∪ 𝑦 ⊆ 𝐴)
10		elpwi 4564	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → 𝑦 ⊆ suc 𝐴)
11	10	unissd 4877	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ ∪ suc 𝐴)
12	1	onunisuci 6477	. . . . . 6 ⊢ ∪ suc 𝐴 = 𝐴
13	11, 12	sseqtrdi 3971	. . . . 5 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ 𝐴)
14	9, 13	nsyl 141	. . . 4 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15		eldif 3909	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16		elpwunsn 4645	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) → suc 𝐴 ∈ 𝑦)
17	15, 16	sylbir 238	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴) → suc 𝐴 ∈ 𝑦)
18	17	ex 418	. . . . 5 ⊢ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴 ∈ 𝑦))
19		df-suc 6361	. . . . . 6 ⊢ suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
20	19	pweqi 4573	. . . . 5 ⊢ 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
21	18, 20	eleq2s 2879	. . . 4 ⊢ (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴 ∈ 𝑦))
22		snelpwi 5412	. . . . 5 ⊢ (suc 𝐴 ∈ 𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23		snfi 9055	. . . . . . . 8 ⊢ {suc 𝐴} ∈ Fin
24	23	jctr 534	. . . . . . 7 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25		elin 3915	. . . . . . 7 ⊢ ({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
26	24, 25	sylibr 237	. . . . . 6 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → {suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin))
27	2	elexi 3473	. . . . . . . 8 ⊢ suc 𝐴 ∈ V
28	27	unisn 4886	. . . . . . 7 ⊢ ∪ {suc 𝐴} = suc 𝐴
29	28	eqcomi 2770	. . . . . 6 ⊢ suc 𝐴 = ∪ {suc 𝐴}
30		unieq 4878	. . . . . . 7 ⊢ (𝑧 = {suc 𝐴} → ∪ 𝑧 = ∪ {suc 𝐴})
31	30	rspceeqv 3599	. . . . . 6 ⊢ (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = ∪ {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
32	26, 29, 31	sylancl 598	. . . . 5 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
33	22, 32	syl 18	. . . 4 ⊢ (suc 𝐴 ∈ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
34	14, 21, 33	syl56 37	. . 3 ⊢ (𝑦 ∈ 𝒫 suc suc 𝐴 → (suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧))
35	34	rgen 3079	. 2 ⊢ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
36	2	onunisuci 6477	. . . 4 ⊢ ∪ suc suc 𝐴 = suc 𝐴
37	36	eqcomi 2770	. . 3 ⊢ suc 𝐴 = ∪ suc suc 𝐴
38	37	iscmp 23686	. 2 ⊢ (suc suc 𝐴 ∈ Comp ↔ (suc suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)))
39	4, 35, 38	mpbir2an 724	1 ⊢ suc suc 𝐴 ∈ Comp

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867  Oncon0 6355  suc csuc 6357  Fincfn 8957  Topctop 23191  Compccmp 23684

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7867  df-1o 8460  df-en 8958  df-fin 8961  df-topgen 17594  df-top 23192  df-bases 23244  df-cmp 23685

This theorem is used by:  onsucsuccmp  37202