Theorem snsspr1 3634

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 3205	. 2 |- { A } C_ ( { A } u. { B } )
2		df-pr 3500	. 2 |- { A , B } = ( { A } u. { B } )
3	1, 2	sseqtr4i 3098	1 |- { A } C_ { A , B }

Syntax hints:   u. cun 3035   C_ wss 3037   {csn 3493   {cpr 3494

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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pr 3500

This theorem is referenced by:  snsstp1  3636  ssprr  3649  uniop  4137  op1stb  4359  op1stbg  4360  ltrelxr  7749