Theorem sssucid 4555

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 3392	. 2 |- A C_ ( A u. { A } )
2		df-suc 4511	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtrri 3283	1 |- A C_ suc A

Syntax hints:   u. cun 3218   C_ wss 3220   {csn 3705   suc csuc 4505

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-suc 4511

This theorem is referenced by:  trsuc  4562  ordsuc  4705  0elnn  4761  sucinc  6708  sucinc2  6709  oasuc  6727  phplem4  7146  phplem4dom  7153  phplem4on  7159  fiintim  7228  fidcenumlemrk  7261  fidcenumlemr  7262  bj-nntrans  16891