Theorem prsstp13 3848

Description: A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.)

Assertion

Ref	Expression
prsstp13	|- { A , C } C_ { A , B , C }

Proof of Theorem prsstp13

Step	Hyp	Ref	Expression
1		prsstp12 3847	. 2 |- { A , C } C_ { A , C , B }
2		tpcomb 3786	. 2 |- { A , B , C } = { A , C , B }
3	1, 2	sseqtrri 3273	1 |- { A , C } C_ { A , B , C }

Syntax hints:   C_ wss 3211   {cpr 3690   {ctp 3691

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 2214

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696  df-tp 3697

This theorem is referenced by:  sstpr  3861