Theorem sucprc 5800

Description: A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)

Assertion

Ref	Expression
sucprc	|- ( -. A e. _V -> suc A = A )

Proof of Theorem sucprc

Step	Hyp	Ref	Expression
1		snprc 4253	. . . 4 |- ( -. A e. _V <-> { A } = (/) )
2	1	biimpi 206	. . 3 |- ( -. A e. _V -> { A } = (/) )
3	2	uneq2d 3767	. 2 |- ( -. A e. _V -> ( A u. { A } ) = ( A u. (/) ) )
4		df-suc 5729	. 2 |- suc A = ( A u. { A } )
5		un0 3967	. . 3 |- ( A u. (/) ) = A
6	5	eqcomi 2631	. 2 |- A = ( A u. (/) )
7	3, 4, 6	3eqtr4g 2681	1 |- ( -. A e. _V -> suc A = A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-suc 5729

This theorem is referenced by:  nsuceq0  5805  sucon  7008  ordsuc  7014  sucprcreg  8506  suc11reg  8516