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

Theorem onsuci 4711
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 4686 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
31, 2ax-mp 8 1  |-  suc  A  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1710   Oncon0 4474   suc csuc 4476
This theorem is referenced by:  1on  6573  2on  6574  3on  6576  4on  6577  tz9.12lem2  7550  tz9.12  7552  rankpwi  7585  bndrank  7603  rankval4  7629  rankxplim3  7641  cfcof  7990  ttukeylem6  8231  onsucconi  25435  onsucsuccmpi  25441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-tr 4195  df-eprel 4387  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-suc 4480
  Copyright terms: Public domain W3C validator