Theorem prsstp13 3712

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 3711	. 2 |- { A , C } C_ { A , C , B }
2		tpcomb 3656	. 2 |- { A , B , C } = { A , C , B }
3	1, 2	sseqtrri 3163	1 |- { A , C } C_ { A , B , C }

Syntax hints:   C_ wss 3102   {cpr 3562   {ctp 3563

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-tp 3569

This theorem is referenced by:  sstpr  3722