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Theorem sstp 3927
Description: The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
sstp  |-  ( A 
C_  { B ,  C ,  D }  <->  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  \/  (
( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) ) )

Proof of Theorem sstp
StepHypRef Expression
1 df-tp 3786 . . 3  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
21sseq2i 3337 . 2  |-  ( A 
C_  { B ,  C ,  D }  <->  A 
C_  ( { B ,  C }  u.  { D } ) )
3 0ss 3620 . . 3  |-  (/)  C_  A
43biantrur 493 . 2  |-  ( A 
C_  ( { B ,  C }  u.  { D } )  <->  ( (/)  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) ) )
5 ssunsn2 3922 . . 3  |-  ( (
(/)  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) )  <->  ( ( (/)  C_  A  /\  A  C_  { B ,  C }
)  \/  ( (
(/)  u.  { D } )  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) ) ) )
63biantrur 493 . . . . 5  |-  ( A 
C_  { B ,  C }  <->  ( (/)  C_  A  /\  A  C_  { B ,  C } ) )
7 sspr 3926 . . . . 5  |-  ( A 
C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
86, 7bitr3i 243 . . . 4  |-  ( (
(/)  C_  A  /\  A  C_ 
{ B ,  C } )  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
9 uncom 3455 . . . . . . . 8  |-  ( (/)  u. 
{ D } )  =  ( { D }  u.  (/) )
10 un0 3616 . . . . . . . 8  |-  ( { D }  u.  (/) )  =  { D }
119, 10eqtri 2428 . . . . . . 7  |-  ( (/)  u. 
{ D } )  =  { D }
1211sseq1i 3336 . . . . . 6  |-  ( (
(/)  u.  { D } )  C_  A  <->  { D }  C_  A
)
13 uncom 3455 . . . . . . 7  |-  ( { B ,  C }  u.  { D } )  =  ( { D }  u.  { B ,  C } )
1413sseq2i 3337 . . . . . 6  |-  ( A 
C_  ( { B ,  C }  u.  { D } )  <->  A  C_  ( { D }  u.  { B ,  C }
) )
1512, 14anbi12i 679 . . . . 5  |-  ( ( ( (/)  u.  { D } )  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) )  <->  ( { D }  C_  A  /\  A  C_  ( { D }  u.  { B ,  C } ) ) )
16 ssunpr 3925 . . . . 5  |-  ( ( { D }  C_  A  /\  A  C_  ( { D }  u.  { B ,  C }
) )  <->  ( ( A  =  { D }  \/  A  =  ( { D }  u.  { B } ) )  \/  ( A  =  ( { D }  u.  { C } )  \/  A  =  ( { D }  u.  { B ,  C }
) ) ) )
17 uncom 3455 . . . . . . . . 9  |-  ( { D }  u.  { B } )  =  ( { B }  u.  { D } )
18 df-pr 3785 . . . . . . . . 9  |-  { B ,  D }  =  ( { B }  u.  { D } )
1917, 18eqtr4i 2431 . . . . . . . 8  |-  ( { D }  u.  { B } )  =  { B ,  D }
2019eqeq2i 2418 . . . . . . 7  |-  ( A  =  ( { D }  u.  { B } )  <->  A  =  { B ,  D }
)
2120orbi2i 506 . . . . . 6  |-  ( ( A  =  { D }  \/  A  =  ( { D }  u.  { B } ) )  <-> 
( A  =  { D }  \/  A  =  { B ,  D } ) )
22 uncom 3455 . . . . . . . . 9  |-  ( { D }  u.  { C } )  =  ( { C }  u.  { D } )
23 df-pr 3785 . . . . . . . . 9  |-  { C ,  D }  =  ( { C }  u.  { D } )
2422, 23eqtr4i 2431 . . . . . . . 8  |-  ( { D }  u.  { C } )  =  { C ,  D }
2524eqeq2i 2418 . . . . . . 7  |-  ( A  =  ( { D }  u.  { C } )  <->  A  =  { C ,  D }
)
261, 13eqtr2i 2429 . . . . . . . 8  |-  ( { D }  u.  { B ,  C }
)  =  { B ,  C ,  D }
2726eqeq2i 2418 . . . . . . 7  |-  ( A  =  ( { D }  u.  { B ,  C } )  <->  A  =  { B ,  C ,  D } )
2825, 27orbi12i 508 . . . . . 6  |-  ( ( A  =  ( { D }  u.  { C } )  \/  A  =  ( { D }  u.  { B ,  C } ) )  <-> 
( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) )
2921, 28orbi12i 508 . . . . 5  |-  ( ( ( A  =  { D }  \/  A  =  ( { D }  u.  { B } ) )  \/  ( A  =  ( { D }  u.  { C } )  \/  A  =  ( { D }  u.  { B ,  C }
) ) )  <->  ( ( A  =  { D }  \/  A  =  { B ,  D }
)  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) )
3015, 16, 293bitri 263 . . . 4  |-  ( ( ( (/)  u.  { D } )  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) )  <->  ( ( A  =  { D }  \/  A  =  { B ,  D }
)  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) )
318, 30orbi12i 508 . . 3  |-  ( ( ( (/)  C_  A  /\  A  C_  { B ,  C } )  \/  (
( (/)  u.  { D } )  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) ) )  <->  ( (
( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  \/  (
( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) ) )
325, 31bitri 241 . 2  |-  ( (
(/)  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) )  <->  ( (
( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  \/  (
( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) ) )
332, 4, 323bitri 263 1  |-  ( A 
C_  { B ,  C ,  D }  <->  ( ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C }
) )  \/  (
( A  =  { D }  \/  A  =  { B ,  D } )  \/  ( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) ) ) )
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   {ctp 3780
This theorem is referenced by:  pwtp  3976
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  df-tp 3786
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