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Theorem onsuci 6461
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 6436 . 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 1756   Oncon0 4731   suc csuc 4733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-tr 4398  df-eprel 4644  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-suc 4737
This theorem is referenced by:  1on  6939  2on  6940  3on  6942  4on  6943  tz9.12lem2  8007  tz9.12  8009  rankpwi  8042  bndrank  8060  rankval4  8086  rankmapu  8097  rankxplim3  8100  cfcof  8455  ttukeylem6  8695  onsucconi  28295  onsucsuccmpi  28301
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