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Theorem onsucconi 31606
 Description: A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
Hypothesis
Ref Expression
onsucconi.1 𝐴 ∈ On
Assertion
Ref Expression
onsucconi suc 𝐴 ∈ Con

Proof of Theorem onsucconi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 onsucconi.1 . . 3 𝐴 ∈ On
2 onsuctop 31602 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ Top)
31, 2ax-mp 5 . 2 suc 𝐴 ∈ Top
4 elin 3758 . . . 4 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)))
5 elsuci 5708 . . . . 5 (𝑥 ∈ suc 𝐴 → (𝑥𝐴𝑥 = 𝐴))
61onunisuci 5758 . . . . . . 7 suc 𝐴 = 𝐴
76eqcomi 2619 . . . . . 6 𝐴 = suc 𝐴
87cldopn 20645 . . . . 5 (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴𝑥) ∈ suc 𝐴)
91onsuci 6930 . . . . . . . . . 10 suc 𝐴 ∈ On
109oneli 5752 . . . . . . . . 9 ((𝐴𝑥) ∈ suc 𝐴 → (𝐴𝑥) ∈ On)
11 elndif 3696 . . . . . . . . . . . 12 (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴𝑥))
12 on0eln0 5697 . . . . . . . . . . . . . 14 ((𝐴𝑥) ∈ On → (∅ ∈ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ∅))
1312biimprd 237 . . . . . . . . . . . . 13 ((𝐴𝑥) ∈ On → ((𝐴𝑥) ≠ ∅ → ∅ ∈ (𝐴𝑥)))
1413necon1bd 2800 . . . . . . . . . . . 12 ((𝐴𝑥) ∈ On → (¬ ∅ ∈ (𝐴𝑥) → (𝐴𝑥) = ∅))
15 ssdif0 3896 . . . . . . . . . . . . 13 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
161onssneli 5754 . . . . . . . . . . . . 13 (𝐴𝑥 → ¬ 𝑥𝐴)
1715, 16sylbir 224 . . . . . . . . . . . 12 ((𝐴𝑥) = ∅ → ¬ 𝑥𝐴)
1811, 14, 17syl56 35 . . . . . . . . . . 11 ((𝐴𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥𝐴))
1918con2d 128 . . . . . . . . . 10 ((𝐴𝑥) ∈ On → (𝑥𝐴 → ¬ ∅ ∈ 𝑥))
201oneli 5752 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ On)
21 on0eln0 5697 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
2221biimprd 237 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2320, 22syl 17 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2423necon1bd 2800 . . . . . . . . . 10 (𝑥𝐴 → (¬ ∅ ∈ 𝑥𝑥 = ∅))
2519, 24sylcom 30 . . . . . . . . 9 ((𝐴𝑥) ∈ On → (𝑥𝐴𝑥 = ∅))
2610, 25syl 17 . . . . . . . 8 ((𝐴𝑥) ∈ suc 𝐴 → (𝑥𝐴𝑥 = ∅))
2726orim1d 880 . . . . . . 7 ((𝐴𝑥) ∈ suc 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
2827impcom 445 . . . . . 6 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29 vex 3176 . . . . . . 7 𝑥 ∈ V
3029elpr 4146 . . . . . 6 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
3128, 30sylibr 223 . . . . 5 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → 𝑥 ∈ {∅, 𝐴})
325, 8, 31syl2an 493 . . . 4 ((𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
334, 32sylbi 206 . . 3 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
3433ssriv 3572 . 2 (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
357iscon2 21027 . 2 (suc 𝐴 ∈ Con ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
363, 34, 35mpbir2an 957 1 suc 𝐴 ∈ Con
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ∪ cuni 4372  Oncon0 5640  suc csuc 5642  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-topgen 15927  df-top 20521  df-bases 20522  df-cld 20633  df-con 21025 This theorem is referenced by:  onsuccon  31607
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