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Theorem onsucconni 32411
Description: A successor ordinal number is a connected topology. (Contributed by Chen-Pang He, 16-Oct-2015.)
Hypothesis
Ref Expression
onsucconni.1 𝐴 ∈ On
Assertion
Ref Expression
onsucconni suc 𝐴 ∈ Conn

Proof of Theorem onsucconni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 onsucconni.1 . . 3 𝐴 ∈ On
2 onsuctop 32407 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ Top)
31, 2ax-mp 5 . 2 suc 𝐴 ∈ Top
4 elin 3788 . . . 4 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)))
5 elsuci 5779 . . . . 5 (𝑥 ∈ suc 𝐴 → (𝑥𝐴𝑥 = 𝐴))
61onunisuci 5829 . . . . . . 7 suc 𝐴 = 𝐴
76eqcomi 2629 . . . . . 6 𝐴 = suc 𝐴
87cldopn 20816 . . . . 5 (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴𝑥) ∈ suc 𝐴)
91onsuci 7023 . . . . . . . . . 10 suc 𝐴 ∈ On
109oneli 5823 . . . . . . . . 9 ((𝐴𝑥) ∈ suc 𝐴 → (𝐴𝑥) ∈ On)
11 elndif 3726 . . . . . . . . . . . 12 (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴𝑥))
12 on0eln0 5768 . . . . . . . . . . . . . 14 ((𝐴𝑥) ∈ On → (∅ ∈ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ∅))
1312biimprd 238 . . . . . . . . . . . . 13 ((𝐴𝑥) ∈ On → ((𝐴𝑥) ≠ ∅ → ∅ ∈ (𝐴𝑥)))
1413necon1bd 2809 . . . . . . . . . . . 12 ((𝐴𝑥) ∈ On → (¬ ∅ ∈ (𝐴𝑥) → (𝐴𝑥) = ∅))
15 ssdif0 3933 . . . . . . . . . . . . 13 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
161onssneli 5825 . . . . . . . . . . . . 13 (𝐴𝑥 → ¬ 𝑥𝐴)
1715, 16sylbir 225 . . . . . . . . . . . 12 ((𝐴𝑥) = ∅ → ¬ 𝑥𝐴)
1811, 14, 17syl56 36 . . . . . . . . . . 11 ((𝐴𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥𝐴))
1918con2d 129 . . . . . . . . . 10 ((𝐴𝑥) ∈ On → (𝑥𝐴 → ¬ ∅ ∈ 𝑥))
201oneli 5823 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ On)
21 on0eln0 5768 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
2221biimprd 238 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2320, 22syl 17 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2423necon1bd 2809 . . . . . . . . . 10 (𝑥𝐴 → (¬ ∅ ∈ 𝑥𝑥 = ∅))
2519, 24sylcom 30 . . . . . . . . 9 ((𝐴𝑥) ∈ On → (𝑥𝐴𝑥 = ∅))
2610, 25syl 17 . . . . . . . 8 ((𝐴𝑥) ∈ suc 𝐴 → (𝑥𝐴𝑥 = ∅))
2726orim1d 883 . . . . . . 7 ((𝐴𝑥) ∈ suc 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
2827impcom 446 . . . . . 6 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29 vex 3198 . . . . . . 7 𝑥 ∈ V
3029elpr 4189 . . . . . 6 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
3128, 30sylibr 224 . . . . 5 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → 𝑥 ∈ {∅, 𝐴})
325, 8, 31syl2an 494 . . . 4 ((𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
334, 32sylbi 207 . . 3 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
3433ssriv 3599 . 2 (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
357isconn2 21198 . 2 (suc 𝐴 ∈ Conn ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
363, 34, 35mpbir2an 954 1 suc 𝐴 ∈ Conn
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 383  wa 384   = wceq 1481  wcel 1988  wne 2791  cdif 3564  cin 3566  wss 3567  c0 3907  {cpr 4170   cuni 4427  Oncon0 5711  suc csuc 5713  cfv 5876  Topctop 20679  Clsdccld 20801  Conncconn 21195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-ord 5714  df-on 5715  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-topgen 16085  df-top 20680  df-bases 20731  df-cld 20804  df-conn 21196
This theorem is referenced by:  onsucconn  32412
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