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Theorem onsuci 6672
Description: The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
onssi.1  |-  A  e.  On
Assertion
Ref Expression
onsuci  |-  suc  A  e.  On

Proof of Theorem onsuci
StepHypRef Expression
1 onssi.1 . 2  |-  A  e.  On
2 suceloni 6647 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
31, 2ax-mp 5 1  |-  suc  A  e.  On
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1819   Oncon0 4887   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893
This theorem is referenced by:  1on  7155  2on  7156  3on  7158  4on  7159  tz9.12lem2  8223  tz9.12  8225  rankpwi  8258  bndrank  8276  rankval4  8302  rankmapu  8313  rankxplim3  8316  cfcof  8671  ttukeylem6  8911  onsucconi  30107  onsucsuccmpi  30113
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