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Theorem ontgsucval 32768
Description: The topology generated from a successor ordinal number is itself. (Contributed by Chen-Pang He, 11-Oct-2015.)
Assertion
Ref Expression
ontgsucval (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴)

Proof of Theorem ontgsucval
StepHypRef Expression
1 suceloni 7160 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ On)
2 ontgval 32767 . . 3 (suc 𝐴 ∈ On → (topGen‘suc 𝐴) = suc suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ On → (topGen‘suc 𝐴) = suc suc 𝐴)
4 eloni 5876 . . . 4 (𝐴 ∈ On → Ord 𝐴)
5 ordunisuc 7179 . . . 4 (Ord 𝐴 suc 𝐴 = 𝐴)
64, 5syl 17 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
7 suceq 5933 . . 3 ( suc 𝐴 = 𝐴 → suc suc 𝐴 = suc 𝐴)
86, 7syl 17 . 2 (𝐴 ∈ On → suc suc 𝐴 = suc 𝐴)
93, 8eqtrd 2805 1 (𝐴 ∈ On → (topGen‘suc 𝐴) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145   cuni 4574  Ord word 5865  Oncon0 5866  suc csuc 5868  cfv 6031  topGenctg 16306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-ord 5869  df-on 5870  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039  df-topgen 16312
This theorem is referenced by:  onsuctop  32769
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