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Theorem sstp 3878
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 3737 . . 3  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
21sseq2i 3289 . 2  |-  ( A 
C_  { B ,  C ,  D }  <->  A 
C_  ( { B ,  C }  u.  { D } ) )
3 0ss 3571 . . 3  |-  (/)  C_  A
43biantrur 492 . 2  |-  ( A 
C_  ( { B ,  C }  u.  { D } )  <->  ( (/)  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) ) )
5 ssunsn2 3873 . . 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 492 . . . . 5  |-  ( A 
C_  { B ,  C }  <->  ( (/)  C_  A  /\  A  C_  { B ,  C } ) )
7 sspr 3877 . . . . 5  |-  ( A 
C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
86, 7bitr3i 242 . . . 4  |-  ( (
(/)  C_  A  /\  A  C_ 
{ B ,  C } )  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
9 uncom 3407 . . . . . . . 8  |-  ( (/)  u. 
{ D } )  =  ( { D }  u.  (/) )
10 un0 3567 . . . . . . . 8  |-  ( { D }  u.  (/) )  =  { D }
119, 10eqtri 2386 . . . . . . 7  |-  ( (/)  u. 
{ D } )  =  { D }
1211sseq1i 3288 . . . . . 6  |-  ( (
(/)  u.  { D } )  C_  A  <->  { D }  C_  A
)
13 uncom 3407 . . . . . . 7  |-  ( { B ,  C }  u.  { D } )  =  ( { D }  u.  { B ,  C } )
1413sseq2i 3289 . . . . . 6  |-  ( A 
C_  ( { B ,  C }  u.  { D } )  <->  A  C_  ( { D }  u.  { B ,  C }
) )
1512, 14anbi12i 678 . . . . 5  |-  ( ( ( (/)  u.  { D } )  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) )  <->  ( { D }  C_  A  /\  A  C_  ( { D }  u.  { B ,  C } ) ) )
16 ssunpr 3876 . . . . 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 3407 . . . . . . . . 9  |-  ( { D }  u.  { B } )  =  ( { B }  u.  { D } )
18 df-pr 3736 . . . . . . . . 9  |-  { B ,  D }  =  ( { B }  u.  { D } )
1917, 18eqtr4i 2389 . . . . . . . 8  |-  ( { D }  u.  { B } )  =  { B ,  D }
2019eqeq2i 2376 . . . . . . 7  |-  ( A  =  ( { D }  u.  { B } )  <->  A  =  { B ,  D }
)
2120orbi2i 505 . . . . . 6  |-  ( ( A  =  { D }  \/  A  =  ( { D }  u.  { B } ) )  <-> 
( A  =  { D }  \/  A  =  { B ,  D } ) )
22 uncom 3407 . . . . . . . . 9  |-  ( { D }  u.  { C } )  =  ( { C }  u.  { D } )
23 df-pr 3736 . . . . . . . . 9  |-  { C ,  D }  =  ( { C }  u.  { D } )
2422, 23eqtr4i 2389 . . . . . . . 8  |-  ( { D }  u.  { C } )  =  { C ,  D }
2524eqeq2i 2376 . . . . . . 7  |-  ( A  =  ( { D }  u.  { C } )  <->  A  =  { C ,  D }
)
261, 13eqtr2i 2387 . . . . . . . 8  |-  ( { D }  u.  { B ,  C }
)  =  { B ,  C ,  D }
2726eqeq2i 2376 . . . . . . 7  |-  ( A  =  ( { D }  u.  { B ,  C } )  <->  A  =  { B ,  C ,  D } )
2825, 27orbi12i 507 . . . . . 6  |-  ( ( A  =  ( { D }  u.  { C } )  \/  A  =  ( { D }  u.  { B ,  C } ) )  <-> 
( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) )
2921, 28orbi12i 507 . . . . 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 262 . . . 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 507 . . 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 240 . 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 262 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 176    \/ wo 357    /\ wa 358    = wceq 1647    u. cun 3236    C_ wss 3238   (/)c0 3543   {csn 3729   {cpr 3730   {ctp 3731
This theorem is referenced by:  pwtp  3926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ral 2633  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-sn 3735  df-pr 3736  df-tp 3737
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