Theorem onsucsuccmpi 32442

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 7038	. . 3 ⊢ suc 𝐴 ∈ On
3		onsuctop 32432	. . 3 ⊢ (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
4	2, 3	ax-mp 5	. 2 ⊢ suc suc 𝐴 ∈ Top
5	1	onirri 5834	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
6	1, 1	onsucssi 7041	. . . . . . 7 ⊢ (𝐴 ∈ 𝐴 ↔ suc 𝐴 ⊆ 𝐴)
7	5, 6	mtbi 312	. . . . . 6 ⊢ ¬ suc 𝐴 ⊆ 𝐴
8		sseq1 3626	. . . . . 6 ⊢ (suc 𝐴 = ∪ 𝑦 → (suc 𝐴 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴))
9	7, 8	mtbii 316	. . . . 5 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ ∪ 𝑦 ⊆ 𝐴)
10		elpwi 4168	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → 𝑦 ⊆ suc 𝐴)
11	10	unissd 4462	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ ∪ suc 𝐴)
12	1	onunisuci 5841	. . . . . 6 ⊢ ∪ suc 𝐴 = 𝐴
13	11, 12	syl6sseq 3651	. . . . 5 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ 𝐴)
14	9, 13	nsyl 135	. . . 4 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15		eldif 3584	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16		elpwunsn 4224	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) → suc 𝐴 ∈ 𝑦)
17	15, 16	sylbir 225	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴) → suc 𝐴 ∈ 𝑦)
18	17	ex 450	. . . . 5 ⊢ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴 ∈ 𝑦))
19		df-suc 5729	. . . . . 6 ⊢ suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
20	19	pweqi 4162	. . . . 5 ⊢ 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
21	18, 20	eleq2s 2719	. . . 4 ⊢ (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴 ∈ 𝑦))
22		snelpwi 4912	. . . . 5 ⊢ (suc 𝐴 ∈ 𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23		snfi 8038	. . . . . . . 8 ⊢ {suc 𝐴} ∈ Fin
24	23	jctr 565	. . . . . . 7 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25		elin 3796	. . . . . . 7 ⊢ ({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
26	24, 25	sylibr 224	. . . . . 6 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → {suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin))
27	2	elexi 3213	. . . . . . . 8 ⊢ suc 𝐴 ∈ V
28	27	unisn 4451	. . . . . . 7 ⊢ ∪ {suc 𝐴} = suc 𝐴
29	28	eqcomi 2631	. . . . . 6 ⊢ suc 𝐴 = ∪ {suc 𝐴}
30		unieq 4444	. . . . . . . 8 ⊢ (𝑧 = {suc 𝐴} → ∪ 𝑧 = ∪ {suc 𝐴})
31	30	eqeq2d 2632	. . . . . . 7 ⊢ (𝑧 = {suc 𝐴} → (suc 𝐴 = ∪ 𝑧 ↔ suc 𝐴 = ∪ {suc 𝐴}))
32	31	rspcev 3309	. . . . . 6 ⊢ (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = ∪ {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
33	26, 29, 32	sylancl 694	. . . . 5 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
34	22, 33	syl 17	. . . 4 ⊢ (suc 𝐴 ∈ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
35	14, 21, 34	syl56 36	. . 3 ⊢ (𝑦 ∈ 𝒫 suc suc 𝐴 → (suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧))
36	35	rgen 2922	. 2 ⊢ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
37	2	onunisuci 5841	. . . 4 ⊢ ∪ suc suc 𝐴 = suc 𝐴
38	37	eqcomi 2631	. . 3 ⊢ suc 𝐴 = ∪ suc suc 𝐴
39	38	iscmp 21191	. 2 ⊢ (suc suc 𝐴 ∈ Comp ↔ (suc suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)))
40	4, 36, 39	mpbir2an 955	1 ⊢ suc suc 𝐴 ∈ Comp

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  Oncon0 5723  suc csuc 5725  Fincfn 7955  Topctop 20698  Compccmp 21189

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  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-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-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-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-om 7066  df-1o 7560  df-en 7956  df-fin 7959  df-topgen 16104  df-top 20699  df-bases 20750  df-cmp 21190

This theorem is referenced by:  onsucsuccmp  32443