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Theorem sspr 3777
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 3319 . . . . 5  |-  ( (/)  u. 
{ B ,  C } )  =  ( { B ,  C }  u.  (/) )
2 un0 3479 . . . . 5  |-  ( { B ,  C }  u.  (/) )  =  { B ,  C }
31, 2eqtri 2303 . . . 4  |-  ( (/)  u. 
{ B ,  C } )  =  { B ,  C }
43sseq2i 3203 . . 3  |-  ( A 
C_  ( (/)  u.  { B ,  C }
)  <->  A  C_  { B ,  C } )
5 0ss 3483 . . . 4  |-  (/)  C_  A
65biantrur 492 . . 3  |-  ( A 
C_  ( (/)  u.  { B ,  C }
)  <->  ( (/)  C_  A  /\  A  C_  ( (/)  u. 
{ B ,  C } ) ) )
74, 6bitr3i 242 . 2  |-  ( A 
C_  { B ,  C }  <->  ( (/)  C_  A  /\  A  C_  ( (/)  u. 
{ B ,  C } ) ) )
8 ssunpr 3776 . 2  |-  ( (
(/)  C_  A  /\  A  C_  ( (/)  u.  { B ,  C } ) )  <-> 
( ( A  =  (/)  \/  A  =  (
(/)  u.  { B } ) )  \/  ( A  =  (
(/)  u.  { C } )  \/  A  =  ( (/)  u.  { B ,  C }
) ) ) )
9 uncom 3319 . . . . . 6  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
10 un0 3479 . . . . . 6  |-  ( { B }  u.  (/) )  =  { B }
119, 10eqtri 2303 . . . . 5  |-  ( (/)  u. 
{ B } )  =  { B }
1211eqeq2i 2293 . . . 4  |-  ( A  =  ( (/)  u.  { B } )  <->  A  =  { B } )
1312orbi2i 505 . . 3  |-  ( ( A  =  (/)  \/  A  =  ( (/)  u.  { B } ) )  <->  ( A  =  (/)  \/  A  =  { B } ) )
14 uncom 3319 . . . . . 6  |-  ( (/)  u. 
{ C } )  =  ( { C }  u.  (/) )
15 un0 3479 . . . . . 6  |-  ( { C }  u.  (/) )  =  { C }
1614, 15eqtri 2303 . . . . 5  |-  ( (/)  u. 
{ C } )  =  { C }
1716eqeq2i 2293 . . . 4  |-  ( A  =  ( (/)  u.  { C } )  <->  A  =  { C } )
183eqeq2i 2293 . . . 4  |-  ( A  =  ( (/)  u.  { B ,  C }
)  <->  A  =  { B ,  C }
)
1917, 18orbi12i 507 . . 3  |-  ( ( A  =  ( (/)  u. 
{ C } )  \/  A  =  (
(/)  u.  { B ,  C } ) )  <-> 
( A  =  { C }  \/  A  =  { B ,  C } ) )
2013, 19orbi12i 507 . 2  |-  ( ( ( A  =  (/)  \/  A  =  ( (/)  u. 
{ B } ) )  \/  ( A  =  ( (/)  u.  { C } )  \/  A  =  ( (/)  u.  { B ,  C }
) ) )  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
217, 8, 203bitri 262 1  |-  ( A 
C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641
This theorem is referenced by:  sstp  3778  pwpr  3823  indistopon  16738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647
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