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Theorem brsset 25643
Description: For sets, the  SSet binary relationship is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypothesis
Ref Expression
brsset.1  |-  B  e. 
_V
Assertion
Ref Expression
brsset  |-  ( A
SSet B  <->  A  C_  B )

Proof of Theorem brsset
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsset 25642 . . 3  |-  Rel  SSet
21brrelexi 4877 . 2  |-  ( A
SSet B  ->  A  e. 
_V )
3 brsset.1 . . 3  |-  B  e. 
_V
43ssex 4307 . 2  |-  ( A 
C_  B  ->  A  e.  _V )
5 breq1 4175 . . 3  |-  ( x  =  A  ->  (
x SSet B  <->  A SSet B ) )
6 sseq1 3329 . . 3  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
7 opex 4387 . . . . . . 7  |-  <. x ,  B >.  e.  _V
87elrn 5069 . . . . . 6  |-  ( <.
x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  E. y  y (  _E  (x)  ( _V  \  _E  ) ) <.
x ,  B >. )
9 vex 2919 . . . . . . . . 9  |-  y  e. 
_V
10 vex 2919 . . . . . . . . 9  |-  x  e. 
_V
119, 10, 3brtxp 25634 . . . . . . . 8  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  _E  x  /\  y ( _V  \  _E  ) B ) )
12 epel 4457 . . . . . . . . 9  |-  ( y  _E  x  <->  y  e.  x )
13 brv 25631 . . . . . . . . . . 11  |-  y _V B
14 brdif 4220 . . . . . . . . . . 11  |-  ( y ( _V  \  _E  ) B  <->  ( y _V B  /\  -.  y  _E  B ) )
1513, 14mpbiran 885 . . . . . . . . . 10  |-  ( y ( _V  \  _E  ) B  <->  -.  y  _E  B )
163epelc 4456 . . . . . . . . . 10  |-  ( y  _E  B  <->  y  e.  B )
1715, 16xchbinx 302 . . . . . . . . 9  |-  ( y ( _V  \  _E  ) B  <->  -.  y  e.  B )
1812, 17anbi12i 679 . . . . . . . 8  |-  ( ( y  _E  x  /\  y ( _V  \  _E  ) B )  <->  ( y  e.  x  /\  -.  y  e.  B ) )
1911, 18bitri 241 . . . . . . 7  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  e.  x  /\  -.  y  e.  B
) )
2019exbii 1589 . . . . . 6  |-  ( E. y  y (  _E 
(x)  ( _V  \  _E  ) ) <. x ,  B >.  <->  E. y ( y  e.  x  /\  -.  y  e.  B )
)
21 exanali 1592 . . . . . 6  |-  ( E. y ( y  e.  x  /\  -.  y  e.  B )  <->  -.  A. y
( y  e.  x  ->  y  e.  B ) )
228, 20, 213bitrri 264 . . . . 5  |-  ( -. 
A. y ( y  e.  x  ->  y  e.  B )  <->  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
2322con1bii 322 . . . 4  |-  ( -. 
<. x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  A. y ( y  e.  x  ->  y  e.  B ) )
24 df-br 4173 . . . . 5  |-  ( x
SSet B  <->  <. x ,  B >.  e.  SSet )
25 df-sset 25611 . . . . . . 7  |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
2625eleq2i 2468 . . . . . 6  |-  ( <.
x ,  B >.  e. 
SSet 
<-> 
<. x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) ) )
2710, 3opelvv 4883 . . . . . . 7  |-  <. x ,  B >.  e.  ( _V  X.  _V )
28 eldif 3290 . . . . . . 7  |-  ( <.
x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )  <->  ( <. x ,  B >.  e.  ( _V  X.  _V )  /\  -.  <. x ,  B >.  e.  ran  (  _E 
(x)  ( _V  \  _E  ) ) ) )
2927, 28mpbiran 885 . . . . . 6  |-  ( <.
x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
3026, 29bitri 241 . . . . 5  |-  ( <.
x ,  B >.  e. 
SSet 
<->  -.  <. x ,  B >.  e.  ran  (  _E 
(x)  ( _V  \  _E  ) ) )
3124, 30bitri 241 . . . 4  |-  ( x
SSet B  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
32 dfss2 3297 . . . 4  |-  ( x 
C_  B  <->  A. y
( y  e.  x  ->  y  e.  B ) )
3323, 31, 323bitr4i 269 . . 3  |-  ( x
SSet B  <->  x  C_  B )
345, 6, 33vtoclbg 2972 . 2  |-  ( A  e.  _V  ->  ( A SSet B  <->  A  C_  B
) )
352, 4, 34pm5.21nii 343 1  |-  ( A
SSet B  <->  A  C_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    e. wcel 1721   _Vcvv 2916    \ cdif 3277    C_ wss 3280   <.cop 3777   class class class wbr 4172    _E cep 4452    X. cxp 4835   ran crn 4838    (x) ctxp 25587   SSetcsset 25589
This theorem is referenced by:  idsset  25644  dfon3  25646
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-eprel 4454  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-1st 6308  df-2nd 6309  df-txp 25609  df-sset 25611
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