ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sucon GIF version

Theorem sucon 4438
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 4437 . 2 ¬ On ∈ V
2 sucprc 4304 . 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 1316  wcel 1465  Vcvv 2660  Oncon0 4255  suc csuc 4257
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator