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Theorem onsucconni 33778
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 33774 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ Top)
31, 2ax-mp 5 . 2 suc 𝐴 ∈ Top
4 elin 4167 . . . 4 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)))
5 elsuci 6250 . . . . 5 (𝑥 ∈ suc 𝐴 → (𝑥𝐴𝑥 = 𝐴))
61onunisuci 6297 . . . . . . 7 suc 𝐴 = 𝐴
76eqcomi 2828 . . . . . 6 𝐴 = suc 𝐴
87cldopn 21631 . . . . 5 (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴𝑥) ∈ suc 𝐴)
91onsuci 7545 . . . . . . . . . 10 suc 𝐴 ∈ On
109oneli 6291 . . . . . . . . 9 ((𝐴𝑥) ∈ suc 𝐴 → (𝐴𝑥) ∈ On)
11 elndif 4103 . . . . . . . . . . . 12 (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴𝑥))
12 on0eln0 6239 . . . . . . . . . . . . . 14 ((𝐴𝑥) ∈ On → (∅ ∈ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ∅))
1312biimprd 250 . . . . . . . . . . . . 13 ((𝐴𝑥) ∈ On → ((𝐴𝑥) ≠ ∅ → ∅ ∈ (𝐴𝑥)))
1413necon1bd 3032 . . . . . . . . . . . 12 ((𝐴𝑥) ∈ On → (¬ ∅ ∈ (𝐴𝑥) → (𝐴𝑥) = ∅))
15 ssdif0 4321 . . . . . . . . . . . . 13 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
161onssneli 6293 . . . . . . . . . . . . 13 (𝐴𝑥 → ¬ 𝑥𝐴)
1715, 16sylbir 237 . . . . . . . . . . . 12 ((𝐴𝑥) = ∅ → ¬ 𝑥𝐴)
1811, 14, 17syl56 36 . . . . . . . . . . 11 ((𝐴𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥𝐴))
1918con2d 136 . . . . . . . . . 10 ((𝐴𝑥) ∈ On → (𝑥𝐴 → ¬ ∅ ∈ 𝑥))
201oneli 6291 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ On)
21 on0eln0 6239 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
2221biimprd 250 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2320, 22syl 17 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2423necon1bd 3032 . . . . . . . . . 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 410 . . . . . 6 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29 vex 3496 . . . . . . 7 𝑥 ∈ V
3029elpr 4582 . . . . . 6 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
3128, 30sylibr 236 . . . . 5 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → 𝑥 ∈ {∅, 𝐴})
325, 8, 31syl2an 597 . . . 4 ((𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
334, 32sylbi 219 . . 3 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
3433ssriv 3969 . 2 (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
357isconn2 22014 . 2 (suc 𝐴 ∈ Conn ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
363, 34, 35mpbir2an 709 1 suc 𝐴 ∈ Conn
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843   = wceq 1531  wcel 2108  wne 3014  cdif 3931  cin 3933  wss 3934  c0 4289  {cpr 4561   cuni 4830  Oncon0 6184  suc csuc 6186  cfv 6348  Topctop 21493  Clsdccld 21616  Conncconn 22011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-topgen 16709  df-top 21494  df-bases 21546  df-cld 21619  df-conn 22012
This theorem is referenced by:  onsucconn  33779
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