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Theorem sspr 3964
Description: The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
sspr  |-  ( A 
C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )

Proof of Theorem sspr
StepHypRef Expression
1 uncom 3493 . . . . 5  |-  ( (/)  u. 
{ B ,  C } )  =  ( { B ,  C }  u.  (/) )
2 un0 3654 . . . . 5  |-  ( { B ,  C }  u.  (/) )  =  { B ,  C }
31, 2eqtri 2458 . . . 4  |-  ( (/)  u. 
{ B ,  C } )  =  { B ,  C }
43sseq2i 3375 . . 3  |-  ( A 
C_  ( (/)  u.  { B ,  C }
)  <->  A  C_  { B ,  C } )
5 0ss 3658 . . . 4  |-  (/)  C_  A
65biantrur 494 . . 3  |-  ( A 
C_  ( (/)  u.  { B ,  C }
)  <->  ( (/)  C_  A  /\  A  C_  ( (/)  u. 
{ B ,  C } ) ) )
74, 6bitr3i 244 . 2  |-  ( A 
C_  { B ,  C }  <->  ( (/)  C_  A  /\  A  C_  ( (/)  u. 
{ B ,  C } ) ) )
8 ssunpr 3963 . 2  |-  ( (
(/)  C_  A  /\  A  C_  ( (/)  u.  { B ,  C } ) )  <-> 
( ( A  =  (/)  \/  A  =  (
(/)  u.  { B } ) )  \/  ( A  =  (
(/)  u.  { C } )  \/  A  =  ( (/)  u.  { B ,  C }
) ) ) )
9 uncom 3493 . . . . . 6  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
10 un0 3654 . . . . . 6  |-  ( { B }  u.  (/) )  =  { B }
119, 10eqtri 2458 . . . . 5  |-  ( (/)  u. 
{ B } )  =  { B }
1211eqeq2i 2448 . . . 4  |-  ( A  =  ( (/)  u.  { B } )  <->  A  =  { B } )
1312orbi2i 507 . . 3  |-  ( ( A  =  (/)  \/  A  =  ( (/)  u.  { B } ) )  <->  ( A  =  (/)  \/  A  =  { B } ) )
14 uncom 3493 . . . . . 6  |-  ( (/)  u. 
{ C } )  =  ( { C }  u.  (/) )
15 un0 3654 . . . . . 6  |-  ( { C }  u.  (/) )  =  { C }
1614, 15eqtri 2458 . . . . 5  |-  ( (/)  u. 
{ C } )  =  { C }
1716eqeq2i 2448 . . . 4  |-  ( A  =  ( (/)  u.  { C } )  <->  A  =  { C } )
183eqeq2i 2448 . . . 4  |-  ( A  =  ( (/)  u.  { B ,  C }
)  <->  A  =  { B ,  C }
)
1917, 18orbi12i 509 . . 3  |-  ( ( A  =  ( (/)  u. 
{ C } )  \/  A  =  (
(/)  u.  { B ,  C } ) )  <-> 
( A  =  { C }  \/  A  =  { B ,  C } ) )
2013, 19orbi12i 509 . 2  |-  ( ( ( A  =  (/)  \/  A  =  ( (/)  u. 
{ B } ) )  \/  ( A  =  ( (/)  u.  { C } )  \/  A  =  ( (/)  u.  { B ,  C }
) ) )  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
217, 8, 203bitri 264 1  |-  ( A 
C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    u. cun 3320    C_ wss 3322   (/)c0 3630   {csn 3816   {cpr 3817
This theorem is referenced by:  sstp  3965  pwpr  4013  indistopon  17070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823
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