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Theorem opth 4436
Description: The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that  C and  D are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
Hypotheses
Ref Expression
opth1.1  |-  A  e. 
_V
opth1.2  |-  B  e. 
_V
Assertion
Ref Expression
opth  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)

Proof of Theorem opth
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . . 4  |-  A  e. 
_V
2 opth1.2 . . . 4  |-  B  e. 
_V
31, 2opth1 4435 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )
41, 2opi1 4431 . . . . . . 7  |-  { A }  e.  <. A ,  B >.
5 id 21 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  D >. )
64, 5syl5eleq 2523 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  <. C ,  D >. )
7 oprcl 4009 . . . . . 6  |-  ( { A }  e.  <. C ,  D >.  ->  ( C  e.  _V  /\  D  e.  _V ) )
86, 7syl 16 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( C  e.  _V  /\  D  e.  _V )
)
98simprd 451 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  D  e.  _V )
103opeq1d 3991 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  B >. )
1110, 5eqtr3d 2471 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  B >.  = 
<. C ,  D >. )
128simpld 447 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  _V )
13 dfopg 3983 . . . . . . . 8  |-  ( ( C  e.  _V  /\  B  e.  _V )  -> 
<. C ,  B >.  =  { { C } ,  { C ,  B } } )
1412, 2, 13sylancl 645 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  B >.  =  { { C } ,  { C ,  B } } )
1511, 14eqtr3d 2471 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  B } } )
16 dfopg 3983 . . . . . . 7  |-  ( ( C  e.  _V  /\  D  e.  _V )  -> 
<. C ,  D >.  =  { { C } ,  { C ,  D } } )
178, 16syl 16 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  D } } )
1815, 17eqtr3d 2471 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { { C } ,  { C ,  B } }  =  { { C } ,  { C ,  D } } )
19 prex 4407 . . . . . 6  |-  { C ,  B }  e.  _V
20 prex 4407 . . . . . 6  |-  { C ,  D }  e.  _V
2119, 20preqr2 3974 . . . . 5  |-  ( { { C } ,  { C ,  B } }  =  { { C } ,  { C ,  D } }  ->  { C ,  B }  =  { C ,  D } )
2218, 21syl 16 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { C ,  B }  =  { C ,  D } )
23 preq2 3885 . . . . . . 7  |-  ( x  =  D  ->  { C ,  x }  =  { C ,  D }
)
2423eqeq2d 2448 . . . . . 6  |-  ( x  =  D  ->  ( { C ,  B }  =  { C ,  x } 
<->  { C ,  B }  =  { C ,  D } ) )
25 eqeq2 2446 . . . . . 6  |-  ( x  =  D  ->  ( B  =  x  <->  B  =  D ) )
2624, 25imbi12d 313 . . . . 5  |-  ( x  =  D  ->  (
( { C ,  B }  =  { C ,  x }  ->  B  =  x )  <-> 
( { C ,  B }  =  { C ,  D }  ->  B  =  D ) ) )
27 vex 2960 . . . . . 6  |-  x  e. 
_V
282, 27preqr2 3974 . . . . 5  |-  ( { C ,  B }  =  { C ,  x }  ->  B  =  x )
2926, 28vtoclg 3012 . . . 4  |-  ( D  e.  _V  ->  ( { C ,  B }  =  { C ,  D }  ->  B  =  D ) )
309, 22, 29sylc 59 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  B  =  D )
313, 30jca 520 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( A  =  C  /\  B  =  D ) )
32 opeq12 3987 . 2  |-  ( ( A  =  C  /\  B  =  D )  -> 
<. A ,  B >.  = 
<. C ,  D >. )
3331, 32impbii 182 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2957   {csn 3815   {cpr 3816   <.cop 3818
This theorem is referenced by:  opthg  4437  otth2  4440  copsexg  4443  copsex4g  4446  opcom  4451  moop2  4452  opelopabsbOLD  4464  ralxpf  5020  cnvcnvsn  5348  funopg  5486  xpopth  6389  eqop  6390  soxp  6460  fnwelem  6462  opiota  6536  xpdom2  7204  xpf1o  7270  unxpdomlem2  7315  unxpdomlem3  7316  xpwdomg  7554  fseqenlem1  7906  iundom2g  8416  eqresr  9013  cnref1o  10608  hashfun  11701  fsumcom2  12559  xpnnenOLD  12810  qredeu  13108  qnumdenbi  13137  crt  13168  prmreclem3  13287  imasaddfnlem  13754  dprd2da  15601  dprd2d2  15603  ucnima  18312  xppreima2  24061  ofpreima  24082  erdszelem9  24886  fprodcom2  25309  brtp  25373  br8  25380  br6  25381  br4  25382  brsegle  26043  f1opr  26427  pellexlem3  26895  pellex  26899  oprabv  28086  opelopab4  28639  dib1dim  31964  diclspsn  31993  dihopelvalcpre  32047  dihmeetlem4preN  32105  dihmeetlem13N  32118  dih1dimatlem  32128  dihatlat  32133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824
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