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Theorem onsucconi 31409
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 31405 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ Top)
31, 2ax-mp 5 . 2 suc 𝐴 ∈ Top
4 elin 3754 . . . 4 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ↔ (𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)))
5 elsuci 5691 . . . . 5 (𝑥 ∈ suc 𝐴 → (𝑥𝐴𝑥 = 𝐴))
61onunisuci 5741 . . . . . . 7 suc 𝐴 = 𝐴
76eqcomi 2615 . . . . . 6 𝐴 = suc 𝐴
87cldopn 20584 . . . . 5 (𝑥 ∈ (Clsd‘suc 𝐴) → (𝐴𝑥) ∈ suc 𝐴)
91onsuci 6904 . . . . . . . . . 10 suc 𝐴 ∈ On
109oneli 5735 . . . . . . . . 9 ((𝐴𝑥) ∈ suc 𝐴 → (𝐴𝑥) ∈ On)
11 elndif 3692 . . . . . . . . . . . 12 (∅ ∈ 𝑥 → ¬ ∅ ∈ (𝐴𝑥))
12 on0eln0 5680 . . . . . . . . . . . . . 14 ((𝐴𝑥) ∈ On → (∅ ∈ (𝐴𝑥) ↔ (𝐴𝑥) ≠ ∅))
1312biimprd 236 . . . . . . . . . . . . 13 ((𝐴𝑥) ∈ On → ((𝐴𝑥) ≠ ∅ → ∅ ∈ (𝐴𝑥)))
1413necon1bd 2796 . . . . . . . . . . . 12 ((𝐴𝑥) ∈ On → (¬ ∅ ∈ (𝐴𝑥) → (𝐴𝑥) = ∅))
15 ssdif0 3892 . . . . . . . . . . . . 13 (𝐴𝑥 ↔ (𝐴𝑥) = ∅)
161onssneli 5737 . . . . . . . . . . . . 13 (𝐴𝑥 → ¬ 𝑥𝐴)
1715, 16sylbir 223 . . . . . . . . . . . 12 ((𝐴𝑥) = ∅ → ¬ 𝑥𝐴)
1811, 14, 17syl56 35 . . . . . . . . . . 11 ((𝐴𝑥) ∈ On → (∅ ∈ 𝑥 → ¬ 𝑥𝐴))
1918con2d 127 . . . . . . . . . 10 ((𝐴𝑥) ∈ On → (𝑥𝐴 → ¬ ∅ ∈ 𝑥))
201oneli 5735 . . . . . . . . . . . 12 (𝑥𝐴𝑥 ∈ On)
21 on0eln0 5680 . . . . . . . . . . . . 13 (𝑥 ∈ On → (∅ ∈ 𝑥𝑥 ≠ ∅))
2221biimprd 236 . . . . . . . . . . . 12 (𝑥 ∈ On → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2320, 22syl 17 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 ≠ ∅ → ∅ ∈ 𝑥))
2423necon1bd 2796 . . . . . . . . . 10 (𝑥𝐴 → (¬ ∅ ∈ 𝑥𝑥 = ∅))
2519, 24sylcom 30 . . . . . . . . 9 ((𝐴𝑥) ∈ On → (𝑥𝐴𝑥 = ∅))
2610, 25syl 17 . . . . . . . 8 ((𝐴𝑥) ∈ suc 𝐴 → (𝑥𝐴𝑥 = ∅))
2726orim1d 879 . . . . . . 7 ((𝐴𝑥) ∈ suc 𝐴 → ((𝑥𝐴𝑥 = 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴)))
2827impcom 444 . . . . . 6 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → (𝑥 = ∅ ∨ 𝑥 = 𝐴))
29 vex 3172 . . . . . . 7 𝑥 ∈ V
3029elpr 4142 . . . . . 6 (𝑥 ∈ {∅, 𝐴} ↔ (𝑥 = ∅ ∨ 𝑥 = 𝐴))
3128, 30sylibr 222 . . . . 5 (((𝑥𝐴𝑥 = 𝐴) ∧ (𝐴𝑥) ∈ suc 𝐴) → 𝑥 ∈ {∅, 𝐴})
325, 8, 31syl2an 492 . . . 4 ((𝑥 ∈ suc 𝐴𝑥 ∈ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
334, 32sylbi 205 . . 3 (𝑥 ∈ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) → 𝑥 ∈ {∅, 𝐴})
3433ssriv 3568 . 2 (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}
357iscon2 20966 . 2 (suc 𝐴 ∈ Con ↔ (suc 𝐴 ∈ Top ∧ (suc 𝐴 ∩ (Clsd‘suc 𝐴)) ⊆ {∅, 𝐴}))
363, 34, 35mpbir2an 956 1 suc 𝐴 ∈ Con
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382   = wceq 1474  wcel 1976  wne 2776  cdif 3533  cin 3535  wss 3536  c0 3870  {cpr 4123   cuni 4363  Oncon0 5623  suc csuc 5625  cfv 5787  Topctop 20456  Clsdccld 20569  Conccon 20963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-ord 5626  df-on 5627  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-fv 5795  df-topgen 15870  df-top 20460  df-bases 20461  df-cld 20572  df-con 20964
This theorem is referenced by:  onsuccon  31410
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