Theorem snsstp1 3514

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 3512	. . 3 |- { A } C_ { A , B }
2		ssun1 3106	. . 3 |- { A , B } C_ ( { A , B } u. { C } )
3	1, 2	sstri 2954	. 2 |- { A } C_ ( { A , B } u. { C } )
4		df-tp 3383	. 2 |- { A , B , C } = ( { A , B } u. { C } )
5	3, 4	sseqtr4i 2978	1 |- { A } C_ { A , B , C }

Syntax hints:   u. cun 2915   C_ wss 2917   {csn 3375   {cpr 3376   {ctp 3377

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pr 3382  df-tp 3383

This theorem is referenced by: (None)