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Theorem opth 4247
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 4246 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )
41, 2opi1 4242 . . . . . . 7  |-  { A }  e.  <. A ,  B >.
5 id 19 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  D >. )
64, 5syl5eleq 2371 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  <. C ,  D >. )
7 oprcl 3822 . . . . . 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 3804 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  B >. )
1110, 5eqtr3d 2319 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  B >.  = 
<. C ,  D >. )
128simpld 445 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  _V )
13 dfopg 3796 . . . . . . . 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 2319 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  B } } )
16 dfopg 3796 . . . . . . 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 2319 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { { C } ,  { C ,  B } }  =  { { C } ,  { C ,  D } } )
19 prex 4219 . . . . . 6  |-  { C ,  B }  e.  _V
20 prex 4219 . . . . . 6  |-  { C ,  D }  e.  _V
2119, 20preqr2 3789 . . . . 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 3709 . . . . . . 7  |-  ( x  =  D  ->  { C ,  x }  =  { C ,  D }
)
2423eqeq2d 2296 . . . . . 6  |-  ( x  =  D  ->  ( { C ,  B }  =  { C ,  x } 
<->  { C ,  B }  =  { C ,  D } ) )
25 eqeq2 2294 . . . . . 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 2793 . . . . . 6  |-  x  e. 
_V
282, 27preqr2 3789 . . . . 5  |-  ( { C ,  B }  =  { C ,  x }  ->  B  =  x )
2926, 28vtoclg 2845 . . . 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 3800 . 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 1625    e. wcel 1686   _Vcvv 2790   {csn 3642   {cpr 3643   <.cop 3645
This theorem is referenced by:  opthg  4248  otth2  4251  copsexg  4254  copsex4g  4257  opcom  4262  moop2  4263  opelopabsbOLD  4275  ralxpf  4832  cnvcnvsn  5152  funopg  5288  xpopth  6163  eqop  6164  soxp  6230  fnwelem  6232  opiota  6292  xpdom2  6959  xpf1o  7025  unxpdomlem2  7070  unxpdomlem3  7071  xpwdomg  7301  fseqenlem1  7653  iundom2g  8164  eqresr  8761  cnref1o  10351  hashfun  11391  fsumcom2  12239  xpnnenOLD  12490  qredeu  12788  qnumdenbi  12817  crt  12848  prmreclem3  12967  imasaddfnlem  13432  dprd2da  15279  dprd2d2  15281  xppreima2  23214  erdszelem9  23732  brtp  24108  br8  24115  br6  24116  br4  24117  elfuns  24456  brsegle  24733  elo  25052  cbcpcp  25173  f1opr  26402  pellexlem3  26927  pellex  26931  opelopab4  28373  dib1dim  31428  diclspsn  31457  dihopelvalcpre  31511  dihmeetlem4preN  31569  dihmeetlem13N  31582  dih1dimatlem  31592  dihatlat  31597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651
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