Theorem prsstp13 3726

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 3725	. 2 |- { A , C } C_ { A , C , B }
2		tpcomb 3670	. 2 |- { A , B , C } = { A , C , B }
3	1, 2	sseqtrri 3176	1 |- { A , C } C_ { A , B , C }

Syntax hints:   C_ wss 3115   {cpr 3576   {ctp 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3or 969  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-sn 3581  df-pr 3582  df-tp 3583

This theorem is referenced by:  sstpr  3736