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Theorem onsucsuccmpi 34181
 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 7552 . . 3 suc 𝐴 ∈ On
3 onsuctop 34171 . . 3 (suc 𝐴 ∈ On → suc suc 𝐴 ∈ Top)
42, 3ax-mp 5 . 2 suc suc 𝐴 ∈ Top
51onirri 6276 . . . . . . 7 ¬ 𝐴𝐴
61, 1onsucssi 7555 . . . . . . 7 (𝐴𝐴 ↔ suc 𝐴𝐴)
75, 6mtbi 325 . . . . . 6 ¬ suc 𝐴𝐴
8 sseq1 3917 . . . . . 6 (suc 𝐴 = 𝑦 → (suc 𝐴𝐴 𝑦𝐴))
97, 8mtbii 329 . . . . 5 (suc 𝐴 = 𝑦 → ¬ 𝑦𝐴)
10 elpwi 4503 . . . . . . 7 (𝑦 ∈ 𝒫 suc 𝐴𝑦 ⊆ suc 𝐴)
1110unissd 4808 . . . . . 6 (𝑦 ∈ 𝒫 suc 𝐴 𝑦 suc 𝐴)
121onunisuci 6283 . . . . . 6 suc 𝐴 = 𝐴
1311, 12sseqtrdi 3942 . . . . 5 (𝑦 ∈ 𝒫 suc 𝐴 𝑦𝐴)
149, 13nsyl 142 . . . 4 (suc 𝐴 = 𝑦 → ¬ 𝑦 ∈ 𝒫 suc 𝐴)
15 eldif 3868 . . . . . . 7 (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) ↔ (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴))
16 elpwunsn 4578 . . . . . . 7 (𝑦 ∈ (𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∖ 𝒫 suc 𝐴) → suc 𝐴𝑦)
1715, 16sylbir 238 . . . . . 6 ((𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) ∧ ¬ 𝑦 ∈ 𝒫 suc 𝐴) → suc 𝐴𝑦)
1817ex 416 . . . . 5 (𝑦 ∈ 𝒫 (suc 𝐴 ∪ {suc 𝐴}) → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
19 df-suc 6175 . . . . . 6 suc suc 𝐴 = (suc 𝐴 ∪ {suc 𝐴})
2019pweqi 4512 . . . . 5 𝒫 suc suc 𝐴 = 𝒫 (suc 𝐴 ∪ {suc 𝐴})
2118, 20eleq2s 2870 . . . 4 (𝑦 ∈ 𝒫 suc suc 𝐴 → (¬ 𝑦 ∈ 𝒫 suc 𝐴 → suc 𝐴𝑦))
22 snelpwi 5305 . . . . 5 (suc 𝐴𝑦 → {suc 𝐴} ∈ 𝒫 𝑦)
23 snfi 8614 . . . . . . . 8 {suc 𝐴} ∈ Fin
2423jctr 528 . . . . . . 7 ({suc 𝐴} ∈ 𝒫 𝑦 → ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
25 elin 3874 . . . . . . 7 ({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ↔ ({suc 𝐴} ∈ 𝒫 𝑦 ∧ {suc 𝐴} ∈ Fin))
2624, 25sylibr 237 . . . . . 6 ({suc 𝐴} ∈ 𝒫 𝑦 → {suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin))
272elexi 3429 . . . . . . . 8 suc 𝐴 ∈ V
2827unisn 4820 . . . . . . 7 {suc 𝐴} = suc 𝐴
2928eqcomi 2767 . . . . . 6 suc 𝐴 = {suc 𝐴}
30 unieq 4809 . . . . . . 7 (𝑧 = {suc 𝐴} → 𝑧 = {suc 𝐴})
3130rspceeqv 3556 . . . . . 6 (({suc 𝐴} ∈ (𝒫 𝑦 ∩ Fin) ∧ suc 𝐴 = {suc 𝐴}) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
3226, 29, 31sylancl 589 . . . . 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 3080 . 2 𝑦 ∈ 𝒫 suc suc 𝐴(suc 𝐴 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)suc 𝐴 = 𝑧)
362onunisuci 6283 . . . 4 suc suc 𝐴 = suc 𝐴
3736eqcomi 2767 . . 3 suc 𝐴 = suc suc 𝐴
3837iscmp 22088 . 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 710 1 suc suc 𝐴 ∈ Comp
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071   ∖ cdif 3855   ∪ cun 3856   ∩ cin 3857   ⊆ wss 3858  𝒫 cpw 4494  {csn 4522  ∪ cuni 4798  Oncon0 6169  suc csuc 6171  Fincfn 8527  Topctop 21593  Compccmp 22086 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-1o 8112  df-en 8528  df-fin 8531  df-topgen 16775  df-top 21594  df-bases 21646  df-cmp 22087 This theorem is referenced by:  onsucsuccmp  34182
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