Theorem snsspr1 3844

Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.)

Assertion

Ref	Expression
snsspr1	|- { A } C_ { A , B }

Proof of Theorem snsspr1

Step	Hyp	Ref	Expression
1		ssun1 3384	. 2 |- { A } C_ ( { A } u. { B } )
2		df-pr 3698	. 2 |- { A , B } = ( { A } u. { B } )
3	1, 2	sseqtrri 3275	1 |- { A } C_ { A , B }

Syntax hints:   u. cun 3211   C_ wss 3213   {csn 3691   {cpr 3692

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pr 3698

This theorem is referenced by:  snsstp1  3846  ssprr  3862  uniop  4374  op1stb  4601  op1stbg  4602  ltrelxr  8336  lspprid1  14576