Theorem sssucid 4170

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 3135	. 2 |- A C_ ( A u. { A } )
2		df-suc 4126	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtr4i 3032	1 |- A C_ suc A

Syntax hints:   u. cun 2971   C_ wss 2973   {csn 3398   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-suc 4126

This theorem is referenced by:  trsuc  4177  ordsuc  4306  0elnn  4358  sucinc  6048  sucinc2  6049  oasuc  6067  phplem4  6341  phplem4dom  6348  phplem4on  6353  bj-nntrans  10746