MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sstp Structured version   Unicode version

Theorem sstp 4191
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 4032 . . 3  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
21sseq2i 3529 . 2  |-  ( A 
C_  { B ,  C ,  D }  <->  A 
C_  ( { B ,  C }  u.  { D } ) )
3 0ss 3814 . . 3  |-  (/)  C_  A
43biantrur 506 . 2  |-  ( A 
C_  ( { B ,  C }  u.  { D } )  <->  ( (/)  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) ) )
5 ssunsn2 4186 . . 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 506 . . . . 5  |-  ( A 
C_  { B ,  C }  <->  ( (/)  C_  A  /\  A  C_  { B ,  C } ) )
7 sspr 4190 . . . . 5  |-  ( A 
C_  { B ,  C }  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
86, 7bitr3i 251 . . . 4  |-  ( (
(/)  C_  A  /\  A  C_ 
{ B ,  C } )  <->  ( ( A  =  (/)  \/  A  =  { B } )  \/  ( A  =  { C }  \/  A  =  { B ,  C } ) ) )
9 uncom 3648 . . . . . . . 8  |-  ( (/)  u. 
{ D } )  =  ( { D }  u.  (/) )
10 un0 3810 . . . . . . . 8  |-  ( { D }  u.  (/) )  =  { D }
119, 10eqtri 2496 . . . . . . 7  |-  ( (/)  u. 
{ D } )  =  { D }
1211sseq1i 3528 . . . . . 6  |-  ( (
(/)  u.  { D } )  C_  A  <->  { D }  C_  A
)
13 uncom 3648 . . . . . . 7  |-  ( { B ,  C }  u.  { D } )  =  ( { D }  u.  { B ,  C } )
1413sseq2i 3529 . . . . . 6  |-  ( A 
C_  ( { B ,  C }  u.  { D } )  <->  A  C_  ( { D }  u.  { B ,  C }
) )
1512, 14anbi12i 697 . . . . 5  |-  ( ( ( (/)  u.  { D } )  C_  A  /\  A  C_  ( { B ,  C }  u.  { D } ) )  <->  ( { D }  C_  A  /\  A  C_  ( { D }  u.  { B ,  C } ) ) )
16 ssunpr 4189 . . . . 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 3648 . . . . . . . . 9  |-  ( { D }  u.  { B } )  =  ( { B }  u.  { D } )
18 df-pr 4030 . . . . . . . . 9  |-  { B ,  D }  =  ( { B }  u.  { D } )
1917, 18eqtr4i 2499 . . . . . . . 8  |-  ( { D }  u.  { B } )  =  { B ,  D }
2019eqeq2i 2485 . . . . . . 7  |-  ( A  =  ( { D }  u.  { B } )  <->  A  =  { B ,  D }
)
2120orbi2i 519 . . . . . 6  |-  ( ( A  =  { D }  \/  A  =  ( { D }  u.  { B } ) )  <-> 
( A  =  { D }  \/  A  =  { B ,  D } ) )
22 uncom 3648 . . . . . . . . 9  |-  ( { D }  u.  { C } )  =  ( { C }  u.  { D } )
23 df-pr 4030 . . . . . . . . 9  |-  { C ,  D }  =  ( { C }  u.  { D } )
2422, 23eqtr4i 2499 . . . . . . . 8  |-  ( { D }  u.  { C } )  =  { C ,  D }
2524eqeq2i 2485 . . . . . . 7  |-  ( A  =  ( { D }  u.  { C } )  <->  A  =  { C ,  D }
)
261, 13eqtr2i 2497 . . . . . . . 8  |-  ( { D }  u.  { B ,  C }
)  =  { B ,  C ,  D }
2726eqeq2i 2485 . . . . . . 7  |-  ( A  =  ( { D }  u.  { B ,  C } )  <->  A  =  { B ,  C ,  D } )
2825, 27orbi12i 521 . . . . . 6  |-  ( ( A  =  ( { D }  u.  { C } )  \/  A  =  ( { D }  u.  { B ,  C } ) )  <-> 
( A  =  { C ,  D }  \/  A  =  { B ,  C ,  D } ) )
2921, 28orbi12i 521 . . . . 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 271 . . . 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 521 . . 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 249 . 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 271 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 setvar class
Syntax hints:    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   {ctp 4031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030  df-tp 4032
This theorem is referenced by:  pwtp  4242
  Copyright terms: Public domain W3C validator