Theorem snsstp1 3730

Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)

Assertion

Ref	Expression
snsstp1	|- { A } C_ { A , B , C }

Proof of Theorem snsstp1

Step	Hyp	Ref	Expression
1		snsspr1 3728	. . 3 |- { A } C_ { A , B }
2		ssun1 3290	. . 3 |- { A , B } C_ ( { A , B } u. { C } )
3	1, 2	sstri 3156	. 2 |- { A } C_ ( { A , B } u. { C } )
4		df-tp 3591	. 2 |- { A , B , C } = ( { A , B } u. { C } )
5	3, 4	sseqtrri 3182	1 |- { A } C_ { A , B , C }

Syntax hints:   u. cun 3119   C_ wss 3121   {csn 3583   {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-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-pr 3590  df-tp 3591

This theorem is referenced by: (None)