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Theorem sspr 3926
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 3455 . . . . 5  |-  ( (/)  u. 
{ B ,  C } )  =  ( { B ,  C }  u.  (/) )
2 un0 3616 . . . . 5  |-  ( { B ,  C }  u.  (/) )  =  { B ,  C }
31, 2eqtri 2428 . . . 4  |-  ( (/)  u. 
{ B ,  C } )  =  { B ,  C }
43sseq2i 3337 . . 3  |-  ( A 
C_  ( (/)  u.  { B ,  C }
)  <->  A  C_  { B ,  C } )
5 0ss 3620 . . . 4  |-  (/)  C_  A
65biantrur 493 . . 3  |-  ( A 
C_  ( (/)  u.  { B ,  C }
)  <->  ( (/)  C_  A  /\  A  C_  ( (/)  u. 
{ B ,  C } ) ) )
74, 6bitr3i 243 . 2  |-  ( A 
C_  { B ,  C }  <->  ( (/)  C_  A  /\  A  C_  ( (/)  u. 
{ B ,  C } ) ) )
8 ssunpr 3925 . 2  |-  ( (
(/)  C_  A  /\  A  C_  ( (/)  u.  { B ,  C } ) )  <-> 
( ( A  =  (/)  \/  A  =  (
(/)  u.  { B } ) )  \/  ( A  =  (
(/)  u.  { C } )  \/  A  =  ( (/)  u.  { B ,  C }
) ) ) )
9 uncom 3455 . . . . . 6  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
10 un0 3616 . . . . . 6  |-  ( { B }  u.  (/) )  =  { B }
119, 10eqtri 2428 . . . . 5  |-  ( (/)  u. 
{ B } )  =  { B }
1211eqeq2i 2418 . . . 4  |-  ( A  =  ( (/)  u.  { B } )  <->  A  =  { B } )
1312orbi2i 506 . . 3  |-  ( ( A  =  (/)  \/  A  =  ( (/)  u.  { B } ) )  <->  ( A  =  (/)  \/  A  =  { B } ) )
14 uncom 3455 . . . . . 6  |-  ( (/)  u. 
{ C } )  =  ( { C }  u.  (/) )
15 un0 3616 . . . . . 6  |-  ( { C }  u.  (/) )  =  { C }
1614, 15eqtri 2428 . . . . 5  |-  ( (/)  u. 
{ C } )  =  { C }
1716eqeq2i 2418 . . . 4  |-  ( A  =  ( (/)  u.  { C } )  <->  A  =  { C } )
183eqeq2i 2418 . . . 4  |-  ( A  =  ( (/)  u.  { B ,  C }
)  <->  A  =  { B ,  C }
)
1917, 18orbi12i 508 . . 3  |-  ( ( A  =  ( (/)  u. 
{ C } )  \/  A  =  (
(/)  u.  { B ,  C } ) )  <-> 
( A  =  { C }  \/  A  =  { B ,  C } ) )
2013, 19orbi12i 508 . 2  |-  ( ( ( A  =  (/)  \/  A  =  ( (/)  u. 
{ B } ) )  \/  ( A  =  ( (/)  u.  { C } )  \/  A  =  ( (/)  u.  { B ,  C }
) ) )  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
217, 8, 203bitri 263 1  |-  ( A 
C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    u. cun 3282    C_ wss 3284   (/)c0 3592   {csn 3778   {cpr 3779
This theorem is referenced by:  sstp  3927  pwpr  3975  indistopon  17024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-sn 3784  df-pr 3785
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