Theorem sssucid 4480

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 3344	. 2 |- A C_ ( A u. { A } )
2		df-suc 4436	. 2 |- suc A = ( A u. { A } )
3	1, 2	sseqtrri 3236	1 |- A C_ suc A

Syntax hints:   u. cun 3172   C_ wss 3174   {csn 3643   suc csuc 4430

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-suc 4436

This theorem is referenced by:  trsuc  4487  ordsuc  4629  0elnn  4685  sucinc  6554  sucinc2  6555  oasuc  6573  phplem4  6977  phplem4dom  6984  phplem4on  6990  fiintim  7054  fidcenumlemrk  7082  fidcenumlemr  7083  bj-nntrans  16086