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Theorem onsucsuccmpi 32567
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.
StepHypRef Expression
1 onsucsuccmpi.1 . . . 4 𝐴 ∈ On
21onsuci 7080 . . 3 suc 𝐴 ∈ On
3 onsuctop 32557 . . 3 (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
42, 3ax-mp 5 . 2 suc suc 𝐴 ∈ Top
51onirri 5872 . . . . . . 7 ¬ 𝐴𝐴
61, 1onsucssi 7083 . . . . . . 7 (𝐴𝐴 ↔ suc 𝐴𝐴)
75, 6mtbi 311 . . . . . 6 ¬ suc 𝐴𝐴
8 sseq1 3659 . . . . . 6 (suc 𝐴 = 𝑦 → (suc 𝐴𝐴 𝑦𝐴))
97, 8mtbii 315 . . . . 5 (suc 𝐴 = 𝑦 → ¬ 𝑦𝐴)
10 elpwi 4201 . . . . . . 7 (𝑦 ∈ 𝒫 suc 𝐴𝑦 ⊆ suc 𝐴)
1110unissd 4494 . . . . . 6 (𝑦 ∈ 𝒫 suc 𝐴 𝑦 suc 𝐴)
121onunisuci 5879 . . . . . 6 suc 𝐴 = 𝐴
1311, 12syl6sseq 3684 . . . . 5 (𝑦 ∈ 𝒫 suc 𝐴 𝑦𝐴)
149, 13nsyl 135 . . . 4 (suc 𝐴 = 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15 eldif 3617 . . . . . . 7 (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16 elpwunsn 4256 . . . . . . 7 (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) → suc 𝐴𝑦)
1715, 16sylbir 225 . . . . . 6 ((𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴) → suc 𝐴𝑦)
1817ex 449 . . . . 5 (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
19 df-suc 5767 . . . . . 6 suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
2019pweqi 4195 . . . . 5 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
2118, 20eleq2s 2748 . . . 4 (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
22 snelpwi 4942 . . . . 5 (suc 𝐴𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23 snfi 8079 . . . . . . . 8 {suc 𝐴} ∈ Fin
2423jctr 564 . . . . . . 7 ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25 elin 3829 . . . . . . 7 ({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
2624, 25sylibr 224 . . . . . 6 ({suc 𝐴} ∈ 𝒫 𝑦 → {suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin))
272elexi 3244 . . . . . . . 8 suc 𝐴 ∈ V
2827unisn 4483 . . . . . . 7 {suc 𝐴} = suc 𝐴
2928eqcomi 2660 . . . . . 6 suc 𝐴 = {suc 𝐴}
30 unieq 4476 . . . . . . . 8 (𝑧 = {suc 𝐴} → 𝑧 = {suc 𝐴})
3130eqeq2d 2661 . . . . . . 7 (𝑧 = {suc 𝐴} → (suc 𝐴 = 𝑧 ↔ suc 𝐴 = {suc 𝐴}))
3231rspcev 3340 . . . . . 6 (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3326, 29, 32sylancl 695 . . . . 5 ({suc 𝐴} ∈ 𝒫 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3422, 33syl 17 . . . 4 (suc 𝐴𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3514, 21, 34syl56 36 . . 3 (𝑦 ∈ 𝒫 suc suc 𝐴 → (suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧))
3635rgen 2951 . 2 𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
372onunisuci 5879 . . . 4 suc suc 𝐴 = suc 𝐴
3837eqcomi 2660 . . 3 suc 𝐴 = suc suc 𝐴
3938iscmp 21239 . 2 (suc suc 𝐴 ∈ Comp ↔ (suc suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)))
404, 36, 39mpbir2an 975 1 suc suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cdif 3604  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468  Oncon0 5761  suc csuc 5763  Fincfn 7997  Topctop 20746  Compccmp 21237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-en 7998  df-fin 8001  df-topgen 16151  df-top 20747  df-bases 20798  df-cmp 21238
This theorem is referenced by:  onsucsuccmp  32568
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