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Theorem ssprr 3556
Description: The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
ssprr  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C } )

Proof of Theorem ssprr
StepHypRef Expression
1 0ss 3289 . . . 4  |-  (/)  C_  { B ,  C }
2 sseq1 3021 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  { B ,  C }  <->  (/)  C_  { B ,  C } ) )
31, 2mpbiri 166 . . 3  |-  ( A  =  (/)  ->  A  C_  { B ,  C }
)
4 snsspr1 3541 . . . 4  |-  { B }  C_  { B ,  C }
5 sseq1 3021 . . . 4  |-  ( A  =  { B }  ->  ( A  C_  { B ,  C }  <->  { B }  C_  { B ,  C } ) )
64, 5mpbiri 166 . . 3  |-  ( A  =  { B }  ->  A  C_  { B ,  C } )
73, 6jaoi 669 . 2  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B ,  C } )
8 snsspr2 3542 . . . 4  |-  { C }  C_  { B ,  C }
9 sseq1 3021 . . . 4  |-  ( A  =  { C }  ->  ( A  C_  { B ,  C }  <->  { C }  C_  { B ,  C } ) )
108, 9mpbiri 166 . . 3  |-  ( A  =  { C }  ->  A  C_  { B ,  C } )
11 eqimss 3052 . . 3  |-  ( A  =  { B ,  C }  ->  A  C_  { B ,  C }
)
1210, 11jaoi 669 . 2  |-  ( ( A  =  { C }  \/  A  =  { B ,  C }
)  ->  A  C_  { B ,  C } )
137, 12jaoi 669 1  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 662    = wceq 1285    C_ wss 2974   (/)c0 3258   {csn 3406   {cpr 3407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pr 3413
This theorem is referenced by:  sstpr  3557  pwprss  3605
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