Theorem prsstp12 3863

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

Assertion

Ref	Expression
prsstp12	|- { A , B } C_ { A , B , C }

Proof of Theorem prsstp12

Step	Hyp	Ref	Expression
1		ssun1 3392	. 2 |- { A , B } C_ ( { A , B } u. { C } )
2		df-tp 3713	. 2 |- { A , B , C } = ( { A , B } u. { C } )
3	1, 2	sseqtrri 3283	1 |- { A , B } C_ { A , B , C }

Syntax hints:   u. cun 3218   C_ wss 3220   {csn 3705   {cpr 3706   {ctp 3707

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-tp 3713

This theorem is referenced by:  prsstp13  3864  prsstp23  3865  sstpr  3877