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Theorem sssnr 3730
Description: Empty set and the singleton itself are subsets of a singleton. Concerning the converse, see exmidsssn 4178. (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 3445 . . 3  |-  (/)  C_  { B }
2 sseq1 3163 . . 3  |-  ( A  =  (/)  ->  ( A 
C_  { B }  <->  (/)  C_ 
{ B } ) )
31, 2mpbiri 167 . 2  |-  ( A  =  (/)  ->  A  C_  { B } )
4 eqimss 3194 . 2  |-  ( A  =  { B }  ->  A  C_  { B } )
53, 4jaoi 706 1  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 698    = wceq 1342    C_ wss 3114   (/)c0 3407   {csn 3573
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-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-dif 3116  df-in 3120  df-ss 3127  df-nul 3408
This theorem is referenced by:  pwsnss  3780
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