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Theorem opth 4245
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 4244 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )
41, 2opi1 4240 . . . . . . 7  |-  { A }  e.  <. A ,  B >.
5 id 19 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  D >. )
64, 5syl5eleq 2369 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  <. C ,  D >. )
7 oprcl 3820 . . . . . 6  |-  ( { A }  e.  <. C ,  D >.  ->  ( C  e.  _V  /\  D  e.  _V ) )
86, 7syl 15 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( C  e.  _V  /\  D  e.  _V )
)
98simprd 449 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  D  e.  _V )
103opeq1d 3802 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  B >. )
1110, 5eqtr3d 2317 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  B >.  = 
<. C ,  D >. )
128simpld 445 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  _V )
13 dfopg 3794 . . . . . . . 8  |-  ( ( C  e.  _V  /\  B  e.  _V )  -> 
<. C ,  B >.  =  { { C } ,  { C ,  B } } )
1412, 2, 13sylancl 643 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  B >.  =  { { C } ,  { C ,  B } } )
1511, 14eqtr3d 2317 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  B } } )
16 dfopg 3794 . . . . . . 7  |-  ( ( C  e.  _V  /\  D  e.  _V )  -> 
<. C ,  D >.  =  { { C } ,  { C ,  D } } )
178, 16syl 15 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  D } } )
1815, 17eqtr3d 2317 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { { C } ,  { C ,  B } }  =  { { C } ,  { C ,  D } } )
19 prex 4217 . . . . . 6  |-  { C ,  B }  e.  _V
20 prex 4217 . . . . . 6  |-  { C ,  D }  e.  _V
2119, 20preqr2 3787 . . . . 5  |-  ( { { C } ,  { C ,  B } }  =  { { C } ,  { C ,  D } }  ->  { C ,  B }  =  { C ,  D } )
2218, 21syl 15 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { C ,  B }  =  { C ,  D } )
23 preq2 3707 . . . . . . 7  |-  ( x  =  D  ->  { C ,  x }  =  { C ,  D }
)
2423eqeq2d 2294 . . . . . 6  |-  ( x  =  D  ->  ( { C ,  B }  =  { C ,  x } 
<->  { C ,  B }  =  { C ,  D } ) )
25 eqeq2 2292 . . . . . 6  |-  ( x  =  D  ->  ( B  =  x  <->  B  =  D ) )
2624, 25imbi12d 311 . . . . 5  |-  ( x  =  D  ->  (
( { C ,  B }  =  { C ,  x }  ->  B  =  x )  <-> 
( { C ,  B }  =  { C ,  D }  ->  B  =  D ) ) )
27 vex 2791 . . . . . 6  |-  x  e. 
_V
282, 27preqr2 3787 . . . . 5  |-  ( { C ,  B }  =  { C ,  x }  ->  B  =  x )
2926, 28vtoclg 2843 . . . 4  |-  ( D  e.  _V  ->  ( { C ,  B }  =  { C ,  D }  ->  B  =  D ) )
309, 22, 29sylc 56 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  B  =  D )
313, 30jca 518 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( A  =  C  /\  B  =  D ) )
32 opeq12 3798 . 2  |-  ( ( A  =  C  /\  B  =  D )  -> 
<. A ,  B >.  = 
<. C ,  D >. )
3331, 32impbii 180 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  <->  ( A  =  C  /\  B  =  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   {cpr 3641   <.cop 3643
This theorem is referenced by:  opthg  4246  otth2  4249  copsexg  4252  copsex4g  4255  opcom  4260  moop2  4261  opelopabsbOLD  4273  ralxpf  4830  cnvcnvsn  5150  funopg  5286  xpopth  6161  eqop  6162  soxp  6228  fnwelem  6230  opiota  6290  xpdom2  6957  xpf1o  7023  unxpdomlem2  7068  unxpdomlem3  7069  xpwdomg  7299  fseqenlem1  7651  iundom2g  8162  eqresr  8759  cnref1o  10349  hashfun  11389  fsumcom2  12237  xpnnenOLD  12488  qredeu  12786  qnumdenbi  12815  crt  12846  prmreclem3  12965  imasaddfnlem  13430  dprd2da  15277  dprd2d2  15279  xppreima2  23212  erdszelem9  23730  brtp  24106  br8  24113  br6  24114  br4  24115  elfuns  24454  brsegle  24731  elo  25041  cbcpcp  25162  f1opr  26391  pellexlem3  26916  pellex  26920  opelopab4  28317  dib1dim  31355  diclspsn  31384  dihopelvalcpre  31438  dihmeetlem4preN  31496  dihmeetlem13N  31509  dih1dimatlem  31519  dihatlat  31524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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