Theorem onsucsuccmpi 36425

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 7858	. . 3 ⊢ suc 𝐴 ∈ On
3		onsuctop 36415	. . 3 ⊢ (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
4	2, 3	ax-mp 5	. 2 ⊢ suc suc 𝐴 ∈ Top
5	1	onirri 6498	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
6	1, 1	onsucssi 7861	. . . . . . 7 ⊢ (𝐴 ∈ 𝐴 ↔ suc 𝐴 ⊆ 𝐴)
7	5, 6	mtbi 322	. . . . . 6 ⊢ ¬ suc 𝐴 ⊆ 𝐴
8		sseq1 4020	. . . . . 6 ⊢ (suc 𝐴 = ∪ 𝑦 → (suc 𝐴 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴))
9	7, 8	mtbii 326	. . . . 5 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ ∪ 𝑦 ⊆ 𝐴)
10		elpwi 4611	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → 𝑦 ⊆ suc 𝐴)
11	10	unissd 4921	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ ∪ suc 𝐴)
12	1	onunisuci 6505	. . . . . 6 ⊢ ∪ suc 𝐴 = 𝐴
13	11, 12	sseqtrdi 4045	. . . . 5 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ 𝐴)
14	9, 13	nsyl 140	. . . 4 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15		eldif 3972	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16		elpwunsn 4688	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) → suc 𝐴 ∈ 𝑦)
17	15, 16	sylbir 235	. . . . . 6 ⊢ ((𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴) → suc 𝐴 ∈ 𝑦)
18	17	ex 412	. . . . 5 ⊢ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴 ∈ 𝑦))
19		df-suc 6391	. . . . . 6 ⊢ suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
20	19	pweqi 4620	. . . . 5 ⊢ 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
21	18, 20	eleq2s 2856	. . . 4 ⊢ (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴 ∈ 𝑦))
22		snelpwi 5453	. . . . 5 ⊢ (suc 𝐴 ∈ 𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23		snfi 9081	. . . . . . . 8 ⊢ {suc 𝐴} ∈ Fin
24	23	jctr 524	. . . . . . 7 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25		elin 3978	. . . . . . 7 ⊢ ({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
26	24, 25	sylibr 234	. . . . . 6 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → {suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin))
27	2	elexi 3500	. . . . . . . 8 ⊢ suc 𝐴 ∈ V
28	27	unisn 4930	. . . . . . 7 ⊢ ∪ {suc 𝐴} = suc 𝐴
29	28	eqcomi 2743	. . . . . 6 ⊢ suc 𝐴 = ∪ {suc 𝐴}
30		unieq 4922	. . . . . . 7 ⊢ (𝑧 = {suc 𝐴} → ∪ 𝑧 = ∪ {suc 𝐴})
31	30	rspceeqv 3644	. . . . . 6 ⊢ (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = ∪ {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
32	26, 29, 31	sylancl 586	. . . . 5 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
33	22, 32	syl 17	. . . 4 ⊢ (suc 𝐴 ∈ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
34	14, 21, 33	syl56 36	. . 3 ⊢ (𝑦 ∈ 𝒫 suc suc 𝐴 → (suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧))
35	34	rgen 3060	. 2 ⊢ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
36	2	onunisuci 6505	. . . 4 ⊢ ∪ suc suc 𝐴 = suc 𝐴
37	36	eqcomi 2743	. . 3 ⊢ suc 𝐴 = ∪ suc suc 𝐴
38	37	iscmp 23411	. 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 711	1 ⊢ suc suc 𝐴 ∈ Comp

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ∖ cdif 3959   ∪ cun 3960   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  {csn 4630  ∪ cuni 4911  Oncon0 6385  suc csuc 6387  Fincfn 8983  Topctop 22914  Compccmp 23409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-en 8984  df-fin 8987  df-topgen 17489  df-top 22915  df-bases 22968  df-cmp 23410

This theorem is referenced by:  onsucsuccmp  36426