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Theorem ssprr 3743
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 3453 . . . 4  |-  (/)  C_  { B ,  C }
2 sseq1 3170 . . . 4  |-  ( A  =  (/)  ->  ( A 
C_  { B ,  C }  <->  (/)  C_  { B ,  C } ) )
31, 2mpbiri 167 . . 3  |-  ( A  =  (/)  ->  A  C_  { B ,  C }
)
4 snsspr1 3728 . . . 4  |-  { B }  C_  { B ,  C }
5 sseq1 3170 . . . 4  |-  ( A  =  { B }  ->  ( A  C_  { B ,  C }  <->  { B }  C_  { B ,  C } ) )
64, 5mpbiri 167 . . 3  |-  ( A  =  { B }  ->  A  C_  { B ,  C } )
73, 6jaoi 711 . 2  |-  ( ( A  =  (/)  \/  A  =  { B } )  ->  A  C_  { B ,  C } )
8 snsspr2 3729 . . . 4  |-  { C }  C_  { B ,  C }
9 sseq1 3170 . . . 4  |-  ( A  =  { C }  ->  ( A  C_  { B ,  C }  <->  { C }  C_  { B ,  C } ) )
108, 9mpbiri 167 . . 3  |-  ( A  =  { C }  ->  A  C_  { B ,  C } )
11 eqimss 3201 . . 3  |-  ( A  =  { B ,  C }  ->  A  C_  { B ,  C }
)
1210, 11jaoi 711 . 2  |-  ( ( A  =  { C }  \/  A  =  { B ,  C }
)  ->  A  C_  { B ,  C } )
137, 12jaoi 711 1  |-  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  ->  A  C_ 
{ B ,  C } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 703    = wceq 1348    C_ wss 3121   (/)c0 3414   {csn 3583   {cpr 3584
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pr 3590
This theorem is referenced by:  sstpr  3744  pwprss  3792
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