Theorem sssucid 4345

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 3244	. 2 |- A C_ ( A u. { A } )
2		df-suc 4301	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtrri 3137	1 |- A C_ suc A

Syntax hints:   u. cun 3074   C_ wss 3076   {csn 3532   suc csuc 4295

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-suc 4301

This theorem is referenced by:  trsuc  4352  ordsuc  4486  0elnn  4540  sucinc  6349  sucinc2  6350  oasuc  6368  phplem4  6757  phplem4dom  6764  phplem4on  6769  fiintim  6825  fidcenumlemrk  6850  fidcenumlemr  6851  bj-nntrans  13320