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Theorem sucon 4537
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 4536 . 2 ¬ On ∈ V
2 sucprc 4397 . 2 (¬ On ∈ V → suc On = On)
31, 2ax-mp 5 1 suc On = On
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1348  wcel 2141  Vcvv 2730  Oncon0 4348  suc csuc 4350
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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356
This theorem is referenced by: (None)
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