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Theorem sucon 7596
Description: The class of all ordinal numbers is its own successor. (Contributed by NM, 12-Sep-2003.)
Assertion
Ref Expression
sucon suc On = On

Proof of Theorem sucon
StepHypRef Expression
1 onprc 7571 . 2 ¬ On ∈ V
2 sucprc 6297 . 2 (¬ On ∈ V → suc On = On)
31, 2ax-mp 5 1 suc On = On
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1543  wcel 2111  Vcvv 3415  Oncon0 6222  suc csuc 6224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-11 2159  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-pss 3894  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4829  df-br 5063  df-opab 5125  df-tr 5171  df-eprel 5469  df-po 5477  df-so 5478  df-fr 5518  df-we 5520  df-ord 6225  df-on 6226  df-suc 6228
This theorem is referenced by:  ordunisuc  7620
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