Theorem sucon 4600

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 4599	. 2 |- -. On e. _V
2		sucprc 4458	. 2 |- ( -. On e. _V -> suc On = On )
3	1, 2	ax-mp 5	1 |- suc On = On

Syntax hints:   -. wn 3   = wceq 1372   e. wcel 2175   _Vcvv 2771   Oncon0 4409   suc csuc 4411

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-uni 3850  df-tr 4142  df-iord 4412  df-on 4414  df-suc 4417

This theorem is referenced by: (None)