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Theorem opthg 4216
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 3758 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21eqeq1d 2174 . . 3  |-  ( x  =  A  ->  ( <. x ,  y >.  =  <. C ,  D >.  <->  <. A ,  y >.  =  <. C ,  D >. ) )
3 eqeq1 2172 . . . 4  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
43anbi1d 461 . . 3  |-  ( x  =  A  ->  (
( x  =  C  /\  y  =  D )  <->  ( A  =  C  /\  y  =  D ) ) )
52, 4bibi12d 234 . 2  |-  ( x  =  A  ->  (
( <. x ,  y
>.  =  <. C ,  D >. 
<->  ( x  =  C  /\  y  =  D ) )  <->  ( <. A ,  y >.  =  <. C ,  D >.  <->  ( A  =  C  /\  y  =  D ) ) ) )
6 opeq2 3759 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
76eqeq1d 2174 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  =  <. C ,  D >.  <->  <. A ,  B >.  = 
<. C ,  D >. ) )
8 eqeq1 2172 . . . 4  |-  ( y  =  B  ->  (
y  =  D  <->  B  =  D ) )
98anbi2d 460 . . 3  |-  ( y  =  B  ->  (
( A  =  C  /\  y  =  D )  <->  ( A  =  C  /\  B  =  D ) ) )
107, 9bibi12d 234 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  =  <. C ,  D >. 
<->  ( A  =  C  /\  y  =  D ) )  <->  ( <. A ,  B >.  =  <. C ,  D >.  <->  ( A  =  C  /\  B  =  D ) ) ) )
11 vex 2729 . . 3  |-  x  e. 
_V
12 vex 2729 . . 3  |-  y  e. 
_V
1311, 12opth 4215 . 2  |-  ( <.
x ,  y >.  =  <. C ,  D >.  <-> 
( x  =  C  /\  y  =  D ) )
145, 10, 13vtocl2g 2790 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 103    <-> wb 104    = wceq 1343    e. wcel 2136   <.cop 3579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585
This theorem is referenced by:  opthg2  4217  xpopth  6144  eqop  6145  inl11  7030  preqlu  7413  cauappcvgprlemladd  7599  elrealeu  7770  qnumdenbi  12124  crth  12156
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