Theorem sssucid 4506

Description: A class is included in its own successor. Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.)

Assertion

Ref	Expression
sssucid	|- A C_ suc A

Proof of Theorem sssucid

Step	Hyp	Ref	Expression
1		ssun1 3367	. 2 |- A C_ ( A u. { A } )
2		df-suc 4462	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtrri 3259	1 |- A C_ suc A

Syntax hints:   u. cun 3195   C_ wss 3197   {csn 3666   suc csuc 4456

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-suc 4462

This theorem is referenced by:  trsuc  4513  ordsuc  4655  0elnn  4711  sucinc  6591  sucinc2  6592  oasuc  6610  phplem4  7016  phplem4dom  7023  phplem4on  7029  fiintim  7093  fidcenumlemrk  7121  fidcenumlemr  7122  bj-nntrans  16314