Theorem sssucid 4433

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 3313	. 2 |- A C_ ( A u. { A } )
2		df-suc 4389	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtrri 3205	1 |- A C_ suc A

Syntax hints:   u. cun 3142   C_ wss 3144   {csn 3607   suc csuc 4383

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-suc 4389

This theorem is referenced by:  trsuc  4440  ordsuc  4580  0elnn  4636  sucinc  6471  sucinc2  6472  oasuc  6490  phplem4  6884  phplem4dom  6891  phplem4on  6896  fiintim  6958  fidcenumlemrk  6984  fidcenumlemr  6985  bj-nntrans  15181