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Theorem brab2a 4903
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 9-Nov-2015.)
Hypotheses
Ref Expression
brab2a.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brab2a.2  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
Assertion
Ref Expression
brab2a  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, D, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)

Proof of Theorem brab2a
StepHypRef Expression
1 simpl 458 . . . . 5  |-  ( ( ( x  e.  C  /\  y  e.  D
)  /\  ph )  -> 
( x  e.  C  /\  y  e.  D
) )
21ssopab2i 4748 . . . 4  |-  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } 
C_  { <. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) }
3 brab2a.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  ph ) }
4 df-xp 4859 . . . 4  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
52, 3, 43sstr4i 3503 . . 3  |-  R  C_  ( C  X.  D
)
65brel 4902 . 2  |-  ( A R B  ->  ( A  e.  C  /\  B  e.  D )
)
7 df-br 4424 . . . 4  |-  ( A R B  <->  <. A ,  B >.  e.  R )
83eleq2i 2499 . . . 4  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
97, 8bitri 252 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) } )
10 brab2a.1 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
1110opelopab2a 4735 . . 3  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( ( x  e.  C  /\  y  e.  D
)  /\  ph ) }  <->  ps ) )
129, 11syl5bb 260 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A R B  <->  ps ) )
136, 12biadan2 646 1  |-  ( A R B  <->  ( ( A  e.  C  /\  B  e.  D )  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   <.cop 4004   class class class wbr 4423   {copab 4481    X. cxp 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859
This theorem is referenced by:  issect  15657  iscgrg  24555  ishlg  24645  iscgra  24849  isinag  24877  isleag  24881
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