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Theorem opth 4203
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
StepHypRef Expression
1 opth1.1 . . . 4  |-  A  e. 
_V
2 opth1.2 . . . 4  |-  B  e. 
_V
31, 2opth1 4202 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )
41, 2opi1 4198 . . . . . . 7  |-  { A }  e.  <. A ,  B >.
5 id 21 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  D >. )
64, 5syl5eleq 2342 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  <. C ,  D >. )
7 oprcl 3780 . . . . . 6  |-  ( { A }  e.  <. C ,  D >.  ->  ( C  e.  _V  /\  D  e.  _V ) )
86, 7syl 17 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( C  e.  _V  /\  D  e.  _V )
)
98simprd 451 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  D  e.  _V )
103opeq1d 3762 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  B >. )
1110, 5eqtr3d 2290 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  B >.  = 
<. C ,  D >. )
128simpld 447 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  _V )
13 dfopg 3754 . . . . . . . 8  |-  ( ( C  e.  _V  /\  B  e.  _V )  -> 
<. C ,  B >.  =  { { C } ,  { C ,  B } } )
1412, 2, 13sylancl 646 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  B >.  =  { { C } ,  { C ,  B } } )
1511, 14eqtr3d 2290 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  B } } )
16 dfopg 3754 . . . . . . 7  |-  ( ( C  e.  _V  /\  D  e.  _V )  -> 
<. C ,  D >.  =  { { C } ,  { C ,  D } } )
178, 16syl 17 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  D } } )
1815, 17eqtr3d 2290 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { { C } ,  { C ,  B } }  =  { { C } ,  { C ,  D } } )
19 prex 4175 . . . . . 6  |-  { C ,  B }  e.  _V
20 prex 4175 . . . . . 6  |-  { C ,  D }  e.  _V
2119, 20preqr2 3747 . . . . 5  |-  ( { { C } ,  { C ,  B } }  =  { { C } ,  { C ,  D } }  ->  { C ,  B }  =  { C ,  D } )
2218, 21syl 17 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { C ,  B }  =  { C ,  D } )
23 preq2 3667 . . . . . . 7  |-  ( x  =  D  ->  { C ,  x }  =  { C ,  D }
)
2423eqeq2d 2267 . . . . . 6  |-  ( x  =  D  ->  ( { C ,  B }  =  { C ,  x } 
<->  { C ,  B }  =  { C ,  D } ) )
25 eqeq2 2265 . . . . . 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 2760 . . . . . 6  |-  x  e. 
_V
282, 27preqr2 3747 . . . . 5  |-  ( { C ,  B }  =  { C ,  x }  ->  B  =  x )
2926, 28vtoclg 2811 . . . 4  |-  ( D  e.  _V  ->  ( { C ,  B }  =  { C ,  D }  ->  B  =  D ) )
309, 22, 29sylc 58 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  B  =  D )
313, 30jca 520 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( A  =  C  /\  B  =  D ) )
32 opeq12 3758 . 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2757   {csn 3600   {cpr 3601   <.cop 3603
This theorem is referenced by:  opthg  4204  otth2  4207  copsexg  4210  copsex4g  4213  opcom  4218  moop2  4219  opelopabsbOLD  4231  ralxpf  4804  cnvcnvsn  5123  funopg  5211  xpopth  6081  eqop  6082  soxp  6148  fnwelem  6150  opiota  6242  xpdom2  6911  xpf1o  6977  unxpdomlem2  7022  unxpdomlem3  7023  xpwdomg  7253  fseqenlem1  7605  iundom2g  8116  eqresr  8713  cnref1o  10302  hashfun  11340  fsumcom2  12188  xpnnenOLD  12436  qredeu  12734  qnumdenbi  12763  crt  12794  prmreclem3  12913  imasaddfnlem  13378  dprd2da  15225  dprd2d2  15227  erdszelem9  23088  brtp  23463  br8  23470  br6  23471  br4  23472  brsegle  24092  elo  24393  cbcpcp  24515  f1opr  25744  pellexlem3  26269  pellex  26273  opelopab4  27354  dib1dim  30506  diclspsn  30535  dihopelvalcpre  30589  dihmeetlem4preN  30647  dihmeetlem13N  30660  dih1dimatlem  30670  dihatlat  30675
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609
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