Theorem sssucid 4253

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 3166	. 2 |- A C_ ( A u. { A } )
2		df-suc 4209	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtr4i 3062	1 |- A C_ suc A

Syntax hints:   u. cun 3000   C_ wss 3002   {csn 3452   suc csuc 4203

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-suc 4209

This theorem is referenced by:  trsuc  4260  ordsuc  4394  0elnn  4447  sucinc  6222  sucinc2  6223  oasuc  6241  phplem4  6627  phplem4dom  6634  phplem4on  6639  fiintim  6695  fidcenumlemrk  6719  fidcenumlemr  6720  bj-nntrans  12150