Theorem snsstp3 3830

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

Assertion

Ref	Expression
snsstp3	|- { C } C_ { A , B , C }

Proof of Theorem snsstp3

Step	Hyp	Ref	Expression
1		ssun2 3373	. 2 |- { C } C_ ( { A , B } u. { C } )
2		df-tp 3681	. 2 |- { A , B , C } = ( { A , B } u. { C } )
3	1, 2	sseqtrri 3263	1 |- { C } C_ { A , B , C }

Syntax hints:   u. cun 3199   C_ wss 3201   {csn 3673   {cpr 3674   {ctp 3675

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-tp 3681

This theorem is referenced by:  sstpr  3845  prdsmulr  13441  mpocnfldmul  14659  cnfldds  14664