Theorem onsucsuccmpi 36656

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 7791	. . 3 ⊢ suc 𝐴 ∈ On
3		onsuctop 36646	. . 3 ⊢ (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
4	2, 3	ax-mp 5	. 2 ⊢ suc suc 𝐴 ∈ Top
5	1	onirri 6439	. . . . . . 7 ⊢ ¬ 𝐴 ∈ 𝐴
6	1, 1	onsucssi 7793	. . . . . . 7 ⊢ (𝐴 ∈ 𝐴 ↔ suc 𝐴 ⊆ 𝐴)
7	5, 6	mtbi 322	. . . . . 6 ⊢ ¬ suc 𝐴 ⊆ 𝐴
8		sseq1 3961	. . . . . 6 ⊢ (suc 𝐴 = ∪ 𝑦 → (suc 𝐴 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴))
9	7, 8	mtbii 326	. . . . 5 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ ∪ 𝑦 ⊆ 𝐴)
10		elpwi 4563	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → 𝑦 ⊆ suc 𝐴)
11	10	unissd 4875	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ ∪ suc 𝐴)
12	1	onunisuci 6446	. . . . . 6 ⊢ ∪ suc 𝐴 = 𝐴
13	11, 12	sseqtrdi 3976	. . . . 5 ⊢ (𝑦 ∈ 𝒫 suc 𝐴 → ∪ 𝑦 ⊆ 𝐴)
14	9, 13	nsyl 140	. . . 4 ⊢ (suc 𝐴 = ∪ 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15		eldif 3913	. . . . . . 7 ⊢ (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16		elpwunsn 4643	. . . . . . 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 6331	. . . . . 6 ⊢ suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
20	19	pweqi 4572	. . . . 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 5399	. . . . 5 ⊢ (suc 𝐴 ∈ 𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23		snfi 8992	. . . . . . . 8 ⊢ {suc 𝐴} ∈ Fin
24	23	jctr 524	. . . . . . 7 ⊢ ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25		elin 3919	. . . . . . 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 3465	. . . . . . . 8 ⊢ suc 𝐴 ∈ V
28	27	unisn 4884	. . . . . . 7 ⊢ ∪ {suc 𝐴} = suc 𝐴
29	28	eqcomi 2746	. . . . . 6 ⊢ suc 𝐴 = ∪ {suc 𝐴}
30		unieq 4876	. . . . . . 7 ⊢ (𝑧 = {suc 𝐴} → ∪ 𝑧 = ∪ {suc 𝐴})
31	30	rspceeqv 3601	. . . . . 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 6446	. . . 4 ⊢ ∪ suc suc 𝐴 = suc 𝐴
37	36	eqcomi 2746	. . 3 ⊢ suc 𝐴 = ∪ suc suc 𝐴
38	37	iscmp 23344	. 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 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  {csn 4582  ∪ cuni 4865  Oncon0 6325  suc csuc 6327  Fincfn 8895  Topctop 22849  Compccmp 23342

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-en 8896  df-fin 8899  df-topgen 17375  df-top 22850  df-bases 22902  df-cmp 23343

This theorem is referenced by:  onsucsuccmp  36657