Theorem sucon 4277

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 4276	. 2 |- -. On e. _V
2		sucprc 4149	. 2 |- ( -. On e. _V -> suc On = On )
3	1, 2	ax-mp 7	1 |- suc On = On

Syntax hints:   -. wn 3   = wceq 1243   e. wcel 1393   _Vcvv 2557   Oncon0 4100   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by: (None)