Theorem prsstp13 3734

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 3733	. 2 |- { A , C } C_ { A , C , B }
2		tpcomb 3678	. 2 |- { A , B , C } = { A , C , B }
3	1, 2	sseqtrri 3182	1 |- { A , C } C_ { A , B , C }

Syntax hints:   C_ wss 3121   {cpr 3584   {ctp 3585

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3or 974  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-tp 3591

This theorem is referenced by:  sstpr  3744