Theorem sucon 7839

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 7813	. 2 ⊢ ¬ On ∈ V
2		sucprc 6471	. 2 ⊢ (¬ On ∈ V → suc On = On)
3	1, 2	ax-mp 5	1 ⊢ suc On = On

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  Vcvv 3488  Oncon0 6395  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-suc 6401

This theorem is referenced by:  ordunisuc  7868