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Theorem brsset 29514
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 29513 . . 3  |-  Rel  SSet
21brrelexi 5030 . 2  |-  ( A
SSet B  ->  A  e. 
_V )
3 brsset.1 . . 3  |-  B  e. 
_V
43ssex 4581 . 2  |-  ( A 
C_  B  ->  A  e.  _V )
5 breq1 4440 . . 3  |-  ( x  =  A  ->  (
x SSet B  <->  A SSet B ) )
6 sseq1 3510 . . 3  |-  ( x  =  A  ->  (
x  C_  B  <->  A  C_  B
) )
7 opex 4701 . . . . . . 7  |-  <. x ,  B >.  e.  _V
87elrn 5233 . . . . . 6  |-  ( <.
x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  E. y  y (  _E  (x)  ( _V  \  _E  ) ) <.
x ,  B >. )
9 vex 3098 . . . . . . . . 9  |-  y  e. 
_V
10 vex 3098 . . . . . . . . 9  |-  x  e. 
_V
119, 10, 3brtxp 29505 . . . . . . . 8  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  _E  x  /\  y ( _V  \  _E  ) B ) )
12 epel 4784 . . . . . . . . 9  |-  ( y  _E  x  <->  y  e.  x )
13 brv 29502 . . . . . . . . . . 11  |-  y _V B
14 brdif 4487 . . . . . . . . . . 11  |-  ( y ( _V  \  _E  ) B  <->  ( y _V B  /\  -.  y  _E  B ) )
1513, 14mpbiran 918 . . . . . . . . . 10  |-  ( y ( _V  \  _E  ) B  <->  -.  y  _E  B )
163epelc 4783 . . . . . . . . . 10  |-  ( y  _E  B  <->  y  e.  B )
1715, 16xchbinx 310 . . . . . . . . 9  |-  ( y ( _V  \  _E  ) B  <->  -.  y  e.  B )
1812, 17anbi12i 697 . . . . . . . 8  |-  ( ( y  _E  x  /\  y ( _V  \  _E  ) B )  <->  ( y  e.  x  /\  -.  y  e.  B ) )
1911, 18bitri 249 . . . . . . 7  |-  ( y (  _E  (x)  ( _V  \  _E  ) )
<. x ,  B >.  <->  (
y  e.  x  /\  -.  y  e.  B
) )
2019exbii 1654 . . . . . 6  |-  ( E. y  y (  _E 
(x)  ( _V  \  _E  ) ) <. x ,  B >.  <->  E. y ( y  e.  x  /\  -.  y  e.  B )
)
21 exanali 1657 . . . . . 6  |-  ( E. y ( y  e.  x  /\  -.  y  e.  B )  <->  -.  A. y
( y  e.  x  ->  y  e.  B ) )
228, 20, 213bitrri 272 . . . . 5  |-  ( -. 
A. y ( y  e.  x  ->  y  e.  B )  <->  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
2322con1bii 331 . . . 4  |-  ( -. 
<. x ,  B >.  e. 
ran  (  _E  (x)  ( _V  \  _E  )
)  <->  A. y ( y  e.  x  ->  y  e.  B ) )
24 df-br 4438 . . . . 5  |-  ( x
SSet B  <->  <. x ,  B >.  e.  SSet )
25 df-sset 29480 . . . . . . 7  |-  SSet  =  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )
2625eleq2i 2521 . . . . . 6  |-  ( <.
x ,  B >.  e. 
SSet 
<-> 
<. x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) ) )
2710, 3opelvv 5036 . . . . . . 7  |-  <. x ,  B >.  e.  ( _V  X.  _V )
28 eldif 3471 . . . . . . 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 918 . . . . . 6  |-  ( <.
x ,  B >.  e.  ( ( _V  X.  _V )  \  ran  (  _E  (x)  ( _V  \  _E  ) ) )  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
3026, 29bitri 249 . . . . 5  |-  ( <.
x ,  B >.  e. 
SSet 
<->  -.  <. x ,  B >.  e.  ran  (  _E 
(x)  ( _V  \  _E  ) ) )
3124, 30bitri 249 . . . 4  |-  ( x
SSet B  <->  -.  <. x ,  B >.  e.  ran  (  _E  (x)  ( _V 
\  _E  ) ) )
32 dfss2 3478 . . . 4  |-  ( x 
C_  B  <->  A. y
( y  e.  x  ->  y  e.  B ) )
3323, 31, 323bitr4i 277 . . 3  |-  ( x
SSet B  <->  x  C_  B )
345, 6, 33vtoclbg 3154 . 2  |-  ( A  e.  _V  ->  ( A SSet B  <->  A  C_  B
) )
352, 4, 34pm5.21nii 353 1  |-  ( A
SSet B  <->  A  C_  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381   E.wex 1599    e. wcel 1804   _Vcvv 3095    \ cdif 3458    C_ wss 3461   <.cop 4020   class class class wbr 4437    _E cep 4779    X. cxp 4987   ran crn 4990    (x) ctxp 29454   SSetcsset 29456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-eprel 4781  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-1st 6785  df-2nd 6786  df-txp 29478  df-sset 29480
This theorem is referenced by:  idsset  29515  dfon3  29517  imagesset  29578
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