Theorem snsspr2 4346

Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.)

Assertion

Ref	Expression
snsspr2	|- { B } C_ { A , B }

Proof of Theorem snsspr2

Step	Hyp	Ref	Expression
1		ssun2 3777	. 2 |- { B } C_ ( { A } u. { B } )
2		df-pr 4180	. 2 |- { A , B } = ( { A } u. { B } )
3	1, 2	sseqtr4i 3638	1 |- { B } C_ { A , B }

Syntax hints:   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pr 4180

This theorem is referenced by:  snsstp2  4348  ord3ex  4856  ltrelxr  10099  2strop  15985  2strop1  15988  phlip  16039  prdsco  16128  ipotset  17157  lsppratlem4  19150  ex-res  27298  subfacp1lem2a  31162  dvh3dim3N  36738  algvsca  37752  corclrcl  37999  gsumpr  42139