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Theorem snsstp2 3639
Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.)
Assertion
Ref Expression
snsstp2  |-  { B }  C_  { A ,  B ,  C }

Proof of Theorem snsstp2
StepHypRef Expression
1 snsspr2 3637 . . 3  |-  { B }  C_  { A ,  B }
2 ssun1 3207 . . 3  |-  { A ,  B }  C_  ( { A ,  B }  u.  { C } )
31, 2sstri 3074 . 2  |-  { B }  C_  ( { A ,  B }  u.  { C } )
4 df-tp 3503 . 2  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
53, 4sseqtrri 3100 1  |-  { B }  C_  { A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    u. cun 3037    C_ wss 3039   {csn 3495   {cpr 3496   {ctp 3497
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pr 3502  df-tp 3503
This theorem is referenced by: (None)
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