Theorem sucon 4619

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 4618	. 2 |- -. On e. _V
2		sucprc 4477	. 2 |- ( -. On e. _V -> suc On = On )
3	1, 2	ax-mp 5	1 |- suc On = On

Syntax hints:   -. wn 3   = wceq 1373   e. wcel 2178   _Vcvv 2776   Oncon0 4428   suc csuc 4430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436

This theorem is referenced by: (None)