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Theorem onsuci 4777
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 4752 . 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 1721   Oncon0 4541   suc csuc 4543
This theorem is referenced by:  1on  6690  2on  6691  3on  6693  4on  6694  tz9.12lem2  7670  tz9.12  7672  rankpwi  7705  bndrank  7723  rankval4  7749  rankxplim3  7761  cfcof  8110  ttukeylem6  8350  onsucconi  26091  onsucsuccmpi  26097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547
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