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Theorem opthg 4003
Description: Ordered pair theorem.  C and  D are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opthg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )

Proof of Theorem opthg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3577 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21eqeq1d 2064 . . 3  |-  ( x  =  A  ->  ( <. x ,  y >.  =  <. C ,  D >.  <->  <. A ,  y >.  =  <. C ,  D >. ) )
3 eqeq1 2062 . . . 4  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
43anbi1d 446 . . 3  |-  ( x  =  A  ->  (
( x  =  C  /\  y  =  D )  <->  ( A  =  C  /\  y  =  D ) ) )
52, 4bibi12d 228 . 2  |-  ( x  =  A  ->  (
( <. x ,  y
>.  =  <. C ,  D >. 
<->  ( x  =  C  /\  y  =  D ) )  <->  ( <. A ,  y >.  =  <. C ,  D >.  <->  ( A  =  C  /\  y  =  D ) ) ) )
6 opeq2 3578 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
76eqeq1d 2064 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  =  <. C ,  D >.  <->  <. A ,  B >.  = 
<. C ,  D >. ) )
8 eqeq1 2062 . . . 4  |-  ( y  =  B  ->  (
y  =  D  <->  B  =  D ) )
98anbi2d 445 . . 3  |-  ( y  =  B  ->  (
( A  =  C  /\  y  =  D )  <->  ( A  =  C  /\  B  =  D ) ) )
107, 9bibi12d 228 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  =  <. C ,  D >. 
<->  ( A  =  C  /\  y  =  D ) )  <->  ( <. A ,  B >.  =  <. C ,  D >.  <->  ( A  =  C  /\  B  =  D ) ) ) )
11 vex 2577 . . 3  |-  x  e. 
_V
12 vex 2577 . . 3  |-  y  e. 
_V
1311, 12opth 4002 . 2  |-  ( <.
x ,  y >.  =  <. C ,  D >.  <-> 
( x  =  C  /\  y  =  D ) )
145, 10, 13vtocl2g 2634 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   <.cop 3406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412
This theorem is referenced by:  opthg2  4004  xpopth  5830  eqop  5831  preqlu  6628  cauappcvgprlemladd  6814  elrealeu  6964
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