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Theorem sssnr 3768
Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4217. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnr  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B } )

Proof of Theorem sssnr
StepHypRef Expression
1 0ss 3476 . . 3  |-  (/)  C_  { B }
2 sseq1 3193 . . 3  |-  ( A  =  (/)  ->  ( A 
C_  { B }  <->  (/)  C_ 
{ B } ) )
31, 2mpbiri 168 . 2  |-  ( A  =  (/)  ->  A  C_  { B } )
4 eqimss 3224 . 2  |-  ( A  =  { B }  ->  A  C_  { B } )
53, 4jaoi 717 1  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 709    = wceq 1364    C_ wss 3144   (/)c0 3437   {csn 3607
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438
This theorem is referenced by:  pwsnss  3818
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