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Theorem onsucconni 35625
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 35621 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ Top)
31, 2ax-mp 5 . 2 suc 𝐴 ∈ Top
4 elin 3963 . . . 4 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)))
5 elsuci 6430 . . . . 5 (𝑥 ∈ suc 𝐴 → (𝑥𝐴𝑥 = 𝐴))
61onunisuci 6483 . . . . . . 7 suc 𝐴 = 𝐴
76eqcomi 2739 . . . . . 6 𝐴 = suc 𝐴
87cldopn 22755 . . . . 5 (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴𝑥) ∈ suc 𝐴)
91onsuci 7829 . . . . . . . . . 10 suc 𝐴 ∈ On
109oneli 6477 . . . . . . . . 9 ((𝐴𝑥) ∈ suc 𝐴 → (𝐴𝑥) ∈ On)
11 elndif 4127 . . . . . . . . . . . 12 (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴𝑥))
12 on0eln0 6419 . . . . . . . . . . . . . 14 ((𝐴𝑥) ∈ On → (∅ ∈ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ∅))
1312biimprd 247 . . . . . . . . . . . . 13 ((𝐴𝑥) ∈ On → ((𝐴𝑥) ≠ ∅ → ∅ ∈ (𝐴𝑥)))
1413necon1bd 2956 . . . . . . . . . . . 12 ((𝐴𝑥) ∈ On → (¬ ∅ ∈ (𝐴𝑥) → (𝐴𝑥) = ∅))
15 ssdif0 4362 . . . . . . . . . . . . 13 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
161onssneli 6479 . . . . . . . . . . . . 13 (𝐴𝑥 → ¬ 𝑥𝐴)
1715, 16sylbir 234 . . . . . . . . . . . 12 ((𝐴𝑥) = ∅ → ¬ 𝑥𝐴)
1811, 14, 17syl56 36 . . . . . . . . . . 11 ((𝐴𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥𝐴))
1918con2d 134 . . . . . . . . . 10 ((𝐴𝑥) ∈ On → (𝑥𝐴 → ¬ ∅ ∈ 𝑥))
201oneli 6477 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ On)
21 on0eln0 6419 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
2221biimprd 247 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2320, 22syl 17 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2423necon1bd 2956 . . . . . . . . . 10 (𝑥𝐴 → (¬ ∅ ∈ 𝑥𝑥 = ∅))
2519, 24sylcom 30 . . . . . . . . 9 ((𝐴𝑥) ∈ On → (𝑥𝐴𝑥 = ∅))
2610, 25syl 17 . . . . . . . 8 ((𝐴𝑥) ∈ suc 𝐴 → (𝑥𝐴𝑥 = ∅))
2726orim1d 962 . . . . . . 7 ((𝐴𝑥) ∈ suc 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
2827impcom 406 . . . . . 6 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29 vex 3476 . . . . . . 7 𝑥 ∈ V
3029elpr 4650 . . . . . 6 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
3128, 30sylibr 233 . . . . 5 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → 𝑥 ∈ {∅, 𝐴})
325, 8, 31syl2an 594 . . . 4 ((𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
334, 32sylbi 216 . . 3 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
3433ssriv 3985 . 2 (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
357isconn2 23138 . 2 (suc 𝐴 ∈ Conn ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
363, 34, 35mpbir2an 707 1 suc 𝐴 ∈ Conn
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 843   = wceq 1539  wcel 2104  wne 2938  cdif 3944  cin 3946  wss 3947  c0 4321  {cpr 4629   cuni 4907  Oncon0 6363  suc csuc 6365  cfv 6542  Topctop 22615  Clsdccld 22740  Conncconn 23135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-topgen 17393  df-top 22616  df-bases 22669  df-cld 22743  df-conn 23136
This theorem is referenced by:  onsucconn  35626
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