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Theorem onsucsuccmpi 32076
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 6986 . . 3 suc 𝐴 ∈ On
3 onsuctop 32066 . . 3 (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
42, 3ax-mp 5 . 2 suc suc 𝐴 ∈ Top
51onirri 5796 . . . . . . 7 ¬ 𝐴𝐴
61, 1onsucssi 6989 . . . . . . 7 (𝐴𝐴 ↔ suc 𝐴𝐴)
75, 6mtbi 312 . . . . . 6 ¬ suc 𝐴𝐴
8 sseq1 3610 . . . . . 6 (suc 𝐴 = 𝑦 → (suc 𝐴𝐴 𝑦𝐴))
97, 8mtbii 316 . . . . 5 (suc 𝐴 = 𝑦 → ¬ 𝑦𝐴)
10 elpwi 4145 . . . . . . 7 (𝑦 ∈ 𝒫 suc 𝐴𝑦 ⊆ suc 𝐴)
1110unissd 4433 . . . . . 6 (𝑦 ∈ 𝒫 suc 𝐴 𝑦 suc 𝐴)
121onunisuci 5803 . . . . . 6 suc 𝐴 = 𝐴
1311, 12syl6sseq 3635 . . . . 5 (𝑦 ∈ 𝒫 suc 𝐴 𝑦𝐴)
149, 13nsyl 135 . . . 4 (suc 𝐴 = 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15 eldif 3570 . . . . . . 7 (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16 elpwunsn 4200 . . . . . . 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 450 . . . . 5 (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
19 df-suc 5691 . . . . . 6 suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
2019pweqi 4139 . . . . 5 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
2118, 20eleq2s 2722 . . . 4 (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
22 snelpwi 4878 . . . . 5 (suc 𝐴𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23 snfi 7983 . . . . . . . 8 {suc 𝐴} ∈ Fin
2423jctr 564 . . . . . . 7 ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25 elin 3779 . . . . . . 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 3204 . . . . . . . 8 suc 𝐴 ∈ V
2827unisn 4422 . . . . . . 7 {suc 𝐴} = suc 𝐴
2928eqcomi 2635 . . . . . 6 suc 𝐴 = {suc 𝐴}
30 unieq 4415 . . . . . . . 8 (𝑧 = {suc 𝐴} → 𝑧 = {suc 𝐴})
3130eqeq2d 2636 . . . . . . 7 (𝑧 = {suc 𝐴} → (suc 𝐴 = 𝑧 ↔ suc 𝐴 = {suc 𝐴}))
3231rspcev 3300 . . . . . 6 (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3326, 29, 32sylancl 693 . . . . 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 2922 . 2 𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
372onunisuci 5803 . . . 4 suc suc 𝐴 = suc 𝐴
3837eqcomi 2635 . . 3 suc 𝐴 = suc suc 𝐴
3938iscmp 21096 . 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 954 1 suc suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cdif 3557  cun 3558  cin 3559  wss 3560  𝒫 cpw 4135  {csn 4153   cuni 4407  Oncon0 5685  suc csuc 5687  Fincfn 7900  Topctop 20612  Compccmp 21094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-en 7901  df-fin 7904  df-topgen 16020  df-top 20616  df-bases 20617  df-cmp 21095
This theorem is referenced by:  onsucsuccmp  32077
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