Theorem onsucsuccmpi 36641

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 7783	. . 3 ⊢ suc 𝐴 ∈ On
3		onsuctop 36631	. . 3 ⊢ (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
4	2, 3	ax-mp 5	. 2 ⊢ suc suc 𝐴 ∈ Top
5	1	onirri 6431	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
6	1, 1	onsucssi 7785	. . . . . . 7 ⊢ (𝐴 ∈ 𝐴 ↔ suc 𝐴 ⊆ 𝐴)
7	5, 6	mtbi 322	. . . . . 6 ⊢ ¬ suc 𝐴 ⊆ 𝐴
8		sseq1 3948	. . . . . 6 ⊢ (suc 𝐴 = ∪ 𝑦 → (suc 𝐴 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴))
9	7, 8	mtbii 326	. . . . 5 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ ∪ 𝑦 ⊆ 𝐴)
10		elpwi 4549	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → 𝑦 ⊆ suc 𝐴)
11	10	unissd 4861	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ ∪ suc 𝐴)
12	1	onunisuci 6438	. . . . . 6 ⊢ ∪ suc 𝐴 = 𝐴
13	11, 12	sseqtrdi 3963	. . . . 5 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ 𝐴)
14	9, 13	nsyl 140	. . . 4 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15		eldif 3900	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16		elpwunsn 4629	. . . . . . 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 6323	. . . . . 6 ⊢ suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
20	19	pweqi 4558	. . . . 5 ⊢ 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
21	18, 20	eleq2s 2855	. . . 4 ⊢ (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴 ∈ 𝑦))
22		snelpwi 5391	. . . . 5 ⊢ (suc 𝐴 ∈ 𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23		snfi 8983	. . . . . . . 8 ⊢ {suc 𝐴} ∈ Fin
24	23	jctr 524	. . . . . . 7 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25		elin 3906	. . . . . . 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 3453	. . . . . . . 8 ⊢ suc 𝐴 ∈ V
28	27	unisn 4870	. . . . . . 7 ⊢ ∪ {suc 𝐴} = suc 𝐴
29	28	eqcomi 2746	. . . . . 6 ⊢ suc 𝐴 = ∪ {suc 𝐴}
30		unieq 4862	. . . . . . 7 ⊢ (𝑧 = {suc 𝐴} → ∪ 𝑧 = ∪ {suc 𝐴})
31	30	rspceeqv 3588	. . . . . 6 ⊢ (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = ∪ {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
32	26, 29, 31	sylancl 587	. . . . 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 3054	. 2 ⊢ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = ∪ 𝑧)
36	2	onunisuci 6438	. . . 4 ⊢ ∪ suc suc 𝐴 = suc 𝐴
37	36	eqcomi 2746	. . 3 ⊢ suc 𝐴 = ∪ suc suc 𝐴
38	37	iscmp 23363	. 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 712	1 ⊢ suc suc 𝐴 ∈ Comp

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851  Oncon0 6317  suc csuc 6319  Fincfn 8886  Topctop 22868  Compccmp 23361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7811  df-1o 8398  df-en 8887  df-fin 8890  df-topgen 17397  df-top 22869  df-bases 22921  df-cmp 23362

This theorem is referenced by:  onsucsuccmp  36642