Theorem sssucid 4462

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 3336	. 2 |- A C_ ( A u. { A } )
2		df-suc 4418	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtrri 3228	1 |- A C_ suc A

Syntax hints:   u. cun 3164   C_ wss 3166   {csn 3633   suc csuc 4412

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-suc 4418

This theorem is referenced by:  trsuc  4469  ordsuc  4611  0elnn  4667  sucinc  6531  sucinc2  6532  oasuc  6550  phplem4  6952  phplem4dom  6959  phplem4on  6964  fiintim  7028  fidcenumlemrk  7056  fidcenumlemr  7057  bj-nntrans  15887