Theorem sssucid 4393

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 3285	. 2 |- A C_ ( A u. { A } )
2		df-suc 4349	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtrri 3177	1 |- A C_ suc A

Syntax hints:   u. cun 3114   C_ wss 3116   {csn 3576   suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-suc 4349

This theorem is referenced by:  trsuc  4400  ordsuc  4540  0elnn  4596  sucinc  6413  sucinc2  6414  oasuc  6432  phplem4  6821  phplem4dom  6828  phplem4on  6833  fiintim  6894  fidcenumlemrk  6919  fidcenumlemr  6920  bj-nntrans  13833