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Theorem onsuci 7831
Description: The successor of an ordinal number is an ordinal number. Inference associated with onsuc 7803 and onsucb 7809. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
onssi.1 𝐴 ∈ On
Assertion
Ref Expression
onsuci suc 𝐴 ∈ On

Proof of Theorem onsuci
StepHypRef Expression
1 onssi.1 . 2 𝐴 ∈ On
2 onsuc 7803 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
31, 2ax-mp 5 1 suc 𝐴 ∈ On
Colors of variables: wff setvar class
Syntax hints:  wcel 2104  Oncon0 6365  suc csuc 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-suc 6371
This theorem is referenced by:  1onOLD  8483  2onOLD  8485  3on  8488  4on  8489  tz9.12lem2  9787  tz9.12  9789  rankpwi  9822  bndrank  9840  rankval4  9866  rankmapu  9877  rankxplim3  9880  cfcof  10273  ttukeylem6  10513  onsucconni  35627  onsucsuccmpi  35633
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