Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  onsuci Unicode version

Theorem onsuci 6673
 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
Assertion
Ref Expression
onsuci

Proof of Theorem onsuci
StepHypRef Expression
1 onssi.1 . 2
2 suceloni 6648 . 2
31, 2ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:  e.wcel 1818   con0 4883  succsuc 4885 This theorem is referenced by:  1on  7156  2on  7157  3on  7159  4on  7160  tz9.12lem2  8227  tz9.12  8229  rankpwi  8262  bndrank  8280  rankval4  8306  rankmapu  8317  rankxplim3  8320  cfcof  8675  ttukeylem6  8915  onsucconi  29902  onsucsuccmpi  29908 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592 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 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  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 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-suc 4889
 Copyright terms: Public domain W3C validator