Theorem sucon 7753

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

Step	Hyp	Ref	Expression
1		onprc 7728	. 2 ⊢ ¬ On ∈ V
2		sucprc 6395	. 2 ⊢ (¬ On ∈ V → suc On = On)
3	1, 2	ax-mp 5	1 ⊢ suc On = On

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119  Vcvv 3432  Oncon0 6317  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by:  ordunisuc  7779