Theorem snsstp3 3774

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 3327	. 2 |- { C } C_ ( { A , B } u. { C } )
2		df-tp 3630	. 2 |- { A , B , C } = ( { A , B } u. { C } )
3	1, 2	sseqtrri 3218	1 |- { C } C_ { A , B , C }

Syntax hints:   u. cun 3155   C_ wss 3157   {csn 3622   {cpr 3623   {ctp 3624

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-tp 3630

This theorem is referenced by:  sstpr  3787  mpocnfldmul  14119  cnfldds  14124