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Theorem onsucsuccmpi 32813
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 7236 . . 3 suc 𝐴 ∈ On
3 onsuctop 32803 . . 3 (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
42, 3ax-mp 5 . 2 suc suc 𝐴 ∈ Top
51onirri 6014 . . . . . . 7 ¬ 𝐴𝐴
61, 1onsucssi 7239 . . . . . . 7 (𝐴𝐴 ↔ suc 𝐴𝐴)
75, 6mtbi 313 . . . . . 6 ¬ suc 𝐴𝐴
8 sseq1 3786 . . . . . 6 (suc 𝐴 = 𝑦 → (suc 𝐴𝐴 𝑦𝐴))
97, 8mtbii 317 . . . . 5 (suc 𝐴 = 𝑦 → ¬ 𝑦𝐴)
10 elpwi 4325 . . . . . . 7 (𝑦 ∈ 𝒫 suc 𝐴𝑦 ⊆ suc 𝐴)
1110unissd 4620 . . . . . 6 (𝑦 ∈ 𝒫 suc 𝐴 𝑦 suc 𝐴)
121onunisuci 6021 . . . . . 6 suc 𝐴 = 𝐴
1311, 12syl6sseq 3811 . . . . 5 (𝑦 ∈ 𝒫 suc 𝐴 𝑦𝐴)
149, 13nsyl 137 . . . 4 (suc 𝐴 = 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15 eldif 3742 . . . . . . 7 (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16 elpwunsn 4381 . . . . . . 7 (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) → suc 𝐴𝑦)
1715, 16sylbir 226 . . . . . 6 ((𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴) → suc 𝐴𝑦)
1817ex 401 . . . . 5 (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
19 df-suc 5914 . . . . . 6 suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
2019pweqi 4319 . . . . 5 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
2118, 20eleq2s 2862 . . . 4 (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
22 snelpwi 5068 . . . . 5 (suc 𝐴𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23 snfi 8245 . . . . . . . 8 {suc 𝐴} ∈ Fin
2423jctr 520 . . . . . . 7 ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25 elin 3958 . . . . . . 7 ({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
2624, 25sylibr 225 . . . . . 6 ({suc 𝐴} ∈ 𝒫 𝑦 → {suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin))
272elexi 3366 . . . . . . . 8 suc 𝐴 ∈ V
2827unisn 4610 . . . . . . 7 {suc 𝐴} = suc 𝐴
2928eqcomi 2774 . . . . . 6 suc 𝐴 = {suc 𝐴}
30 unieq 4602 . . . . . . 7 (𝑧 = {suc 𝐴} → 𝑧 = {suc 𝐴})
3130rspceeqv 3479 . . . . . 6 (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3226, 29, 31sylancl 580 . . . . 5 ({suc 𝐴} ∈ 𝒫 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3322, 32syl 17 . . . 4 (suc 𝐴𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3414, 21, 33syl56 36 . . 3 (𝑦 ∈ 𝒫 suc suc 𝐴 → (suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧))
3534rgen 3069 . 2 𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
362onunisuci 6021 . . . 4 suc suc 𝐴 = suc 𝐴
3736eqcomi 2774 . . 3 suc 𝐴 = suc suc 𝐴
3837iscmp 21471 . 2 (suc suc 𝐴 ∈ Comp ↔ (suc suc 𝐴 ∈ Top ∧ ∀𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)))
394, 35, 38mpbir2an 702 1 suc suc 𝐴 ∈ Comp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  cdif 3729  cun 3730  cin 3731  wss 3732  𝒫 cpw 4315  {csn 4334   cuni 4594  Oncon0 5908  suc csuc 5910  Fincfn 8160  Topctop 20977  Compccmp 21469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-en 8161  df-fin 8164  df-topgen 16372  df-top 20978  df-bases 21030  df-cmp 21470
This theorem is referenced by:  onsucsuccmp  32814
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