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Theorem opthg 4428
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 3976 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21eqeq1d 2443 . . 3  |-  ( x  =  A  ->  ( <. x ,  y >.  =  <. C ,  D >.  <->  <. A ,  y >.  =  <. C ,  D >. ) )
3 eqeq1 2441 . . . 4  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
43anbi1d 686 . . 3  |-  ( x  =  A  ->  (
( x  =  C  /\  y  =  D )  <->  ( A  =  C  /\  y  =  D ) ) )
52, 4bibi12d 313 . 2  |-  ( x  =  A  ->  (
( <. x ,  y
>.  =  <. C ,  D >. 
<->  ( x  =  C  /\  y  =  D ) )  <->  ( <. A ,  y >.  =  <. C ,  D >.  <->  ( A  =  C  /\  y  =  D ) ) ) )
6 opeq2 3977 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
76eqeq1d 2443 . . 3  |-  ( y  =  B  ->  ( <. A ,  y >.  =  <. C ,  D >.  <->  <. A ,  B >.  = 
<. C ,  D >. ) )
8 eqeq1 2441 . . . 4  |-  ( y  =  B  ->  (
y  =  D  <->  B  =  D ) )
98anbi2d 685 . . 3  |-  ( y  =  B  ->  (
( A  =  C  /\  y  =  D )  <->  ( A  =  C  /\  B  =  D ) ) )
107, 9bibi12d 313 . 2  |-  ( y  =  B  ->  (
( <. A ,  y
>.  =  <. C ,  D >. 
<->  ( A  =  C  /\  y  =  D ) )  <->  ( <. A ,  B >.  =  <. C ,  D >.  <->  ( A  =  C  /\  B  =  D ) ) ) )
11 vex 2951 . . 3  |-  x  e. 
_V
12 vex 2951 . . 3  |-  y  e. 
_V
1311, 12opth 4427 . 2  |-  ( <.
x ,  y >.  =  <. C ,  D >.  <-> 
( x  =  C  /\  y  =  D ) )
145, 10, 13vtocl2g 3007 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809
This theorem is referenced by:  opthg2  4429  oteqex  4441  s111  11752  frgpnabllem1  15474  frgpnabllem2  15475  otthg  28018  el2wlkonotot0  28256  dvheveccl  31811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
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