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Theorem opthg 4712
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 4202 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21eqeq1d 2445 . . 3  |-  ( x  =  A  ->  ( <. x ,  y >.  =  <. C ,  D >.  <->  <. A ,  y >.  =  <. C ,  D >. ) )
3 eqeq1 2447 . . . 4  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
43anbi1d 704 . . 3  |-  ( x  =  A  ->  (
( x  =  C  /\  y  =  D )  <->  ( A  =  C  /\  y  =  D ) ) )
52, 4bibi12d 321 . 2  |-  ( x  =  A  ->  (
( <. x ,  y
>.  =  <. C ,  D >. 
<->  ( x  =  C  /\  y  =  D ) )  <->  ( <. A ,  y >.  =  <. C ,  D >.  <->  ( A  =  C  /\  y  =  D ) ) ) )
6 opeq2 4203 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
76eqeq1d 2445 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  =  <. C ,  D >.  <->  <. A ,  B >.  = 
<. C ,  D >. ) )
8 eqeq1 2447 . . . 4  |-  ( y  =  B  ->  (
y  =  D  <->  B  =  D ) )
98anbi2d 703 . . 3  |-  ( y  =  B  ->  (
( A  =  C  /\  y  =  D )  <->  ( A  =  C  /\  B  =  D ) ) )
107, 9bibi12d 321 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  =  <. C ,  D >. 
<->  ( A  =  C  /\  y  =  D ) )  <->  ( <. A ,  B >.  =  <. C ,  D >.  <->  ( A  =  C  /\  B  =  D ) ) ) )
11 vex 3098 . . 3  |-  x  e. 
_V
12 vex 3098 . . 3  |-  y  e. 
_V
1311, 12opth 4711 . 2  |-  ( <.
x ,  y >.  =  <. C ,  D >.  <-> 
( x  =  C  /\  y  =  D ) )
145, 10, 13vtocl2g 3157 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  =  <. C ,  D >.  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804   <.cop 4020
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-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
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-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  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
This theorem is referenced by:  opth1g  4713  opthg2  4714  opthneg  4716  otthg  4720  oteqex  4730  s111  12604  symg2bas  16401  frgpnabllem1  16855  frgpnabllem2  16856  mat1dimbas  18951  el2wlkonotot0  24848  dvheveccl  36579
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