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Theorem onsuci 4438
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 4423 . 2  |-  ( A  e.  On  ->  suc  A  e.  On )
31, 2ax-mp 5 1  |-  suc  A  e.  On
Colors of variables: wff set class
Syntax hints:    e. wcel 1481   Oncon0 4291   suc csuc 4293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-uni 3743  df-tr 4033  df-iord 4294  df-on 4296  df-suc 4299
This theorem is referenced by:  ordtri2orexmid  4444  onsucsssucexmid  4448  ordsoexmid  4483  ordtri2or2exmid  4492  tfr0dm  6225  1on  6326  2on  6328  3on  6330  4on  6331  prarloclemarch2  7249
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