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Theorem brub 28124
Description: Binary relationship form of the upper bound functor. (Contributed by Scott Fenton, 3-May-2018.)
Hypotheses
Ref Expression
brub.1  |-  S  e. 
_V
brub.2  |-  A  e. 
_V
Assertion
Ref Expression
brub  |-  ( SUB R A  <->  A. x  e.  S  x R A )
Distinct variable groups:    x, A    x, R    x, S

Proof of Theorem brub
StepHypRef Expression
1 brub.1 . . . . 5  |-  S  e. 
_V
2 brub.2 . . . . 5  |-  A  e. 
_V
3 brxp 4973 . . . . 5  |-  ( S ( _V  X.  _V ) A  <->  ( S  e. 
_V  /\  A  e.  _V ) )
41, 2, 3mpbir2an 911 . . . 4  |-  S ( _V  X.  _V ) A
5 brdif 4445 . . . 4  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  ( S
( _V  X.  _V ) A  /\  -.  S
( ( _V  \  R )  o.  `'  _E  ) A ) )
64, 5mpbiran 909 . . 3  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  S
( ( _V  \  R )  o.  `'  _E  ) A )
71, 2coepr 27701 . . 3  |-  ( S ( ( _V  \  R )  o.  `'  _E  ) A  <->  E. x  e.  S  x ( _V  \  R ) A )
86, 7xchbinx 310 . 2  |-  ( S ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) ) A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
9 df-ub 28045 . . 3  |- UB R  =  ( ( _V  X.  _V )  \  (
( _V  \  R
)  o.  `'  _E  ) )
109breqi 4401 . 2  |-  ( SUB R A  <->  S (
( _V  X.  _V )  \  ( ( _V 
\  R )  o.  `'  _E  ) ) A )
11 brv 28047 . . . . . 6  |-  x _V A
12 brdif 4445 . . . . . 6  |-  ( x ( _V  \  R
) A  <->  ( x _V A  /\  -.  x R A ) )
1311, 12mpbiran 909 . . . . 5  |-  ( x ( _V  \  R
) A  <->  -.  x R A )
1413rexbii 2861 . . . 4  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  E. x  e.  S  -.  x R A )
15 rexnal 2849 . . . 4  |-  ( E. x  e.  S  -.  x R A  <->  -.  A. x  e.  S  x R A )
1614, 15bitri 249 . . 3  |-  ( E. x  e.  S  x ( _V  \  R
) A  <->  -.  A. x  e.  S  x R A )
1716con2bii 332 . 2  |-  ( A. x  e.  S  x R A  <->  -.  E. x  e.  S  x ( _V  \  R ) A )
188, 10, 173bitr4i 277 1  |-  ( SUB R A  <->  A. x  e.  S  x R A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    e. wcel 1758   A.wral 2796   E.wrex 2797   _Vcvv 3072    \ cdif 3428   class class class wbr 4395    _E cep 4733    X. cxp 4941   `'ccnv 4942    o. ccom 4947  UBcub 28021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-br 4396  df-opab 4454  df-eprel 4735  df-xp 4949  df-cnv 4951  df-co 4952  df-ub 28045
This theorem is referenced by:  brlb  28125
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