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Theorem opth1 4244
Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opth1.1  |-  A  e. 
_V
opth1.2  |-  B  e. 
_V
Assertion
Ref Expression
opth1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )

Proof of Theorem opth1
StepHypRef Expression
1 opth1.1 . . . 4  |-  A  e. 
_V
21sneqr 3780 . . 3  |-  ( { A }  =  { C }  ->  A  =  C )
32a1i 10 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C }  ->  A  =  C ) )
4 opth1.2 . . . . . . . . 9  |-  B  e. 
_V
51, 4opi1 4240 . . . . . . . 8  |-  { A }  e.  <. A ,  B >.
6 id 19 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  D >. )
75, 6syl5eleq 2369 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  <. C ,  D >. )
8 oprcl 3820 . . . . . . 7  |-  ( { A }  e.  <. C ,  D >.  ->  ( C  e.  _V  /\  D  e.  _V ) )
97, 8syl 15 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( C  e.  _V  /\  D  e.  _V )
)
109simpld 445 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  _V )
11 prid1g 3732 . . . . 5  |-  ( C  e.  _V  ->  C  e.  { C ,  D } )
1210, 11syl 15 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  { C ,  D } )
13 eleq2 2344 . . . 4  |-  ( { A }  =  { C ,  D }  ->  ( C  e.  { A }  <->  C  e.  { C ,  D } ) )
1412, 13syl5ibrcom 213 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C ,  D }  ->  C  e.  { A } ) )
15 elsni 3664 . . . 4  |-  ( C  e.  { A }  ->  C  =  A )
1615eqcomd 2288 . . 3  |-  ( C  e.  { A }  ->  A  =  C )
1714, 16syl6 29 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C ,  D }  ->  A  =  C ) )
18 dfopg 3794 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  -> 
<. C ,  D >.  =  { { C } ,  { C ,  D } } )
197, 8, 183syl 18 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  D } } )
207, 19eleqtrd 2359 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  { { C } ,  { C ,  D } } )
21 elpri 3660 . . 3  |-  ( { A }  e.  { { C } ,  { C ,  D } }  ->  ( { A }  =  { C }  \/  { A }  =  { C ,  D } ) )
2220, 21syl 15 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C }  \/  { A }  =  { C ,  D }
) )
233, 17, 22mpjaod 370 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   {cpr 3641   <.cop 3643
This theorem is referenced by:  opth  4245  dmsnopg  5144  funcnvsn  5297  oprabid  5882  seqomlem2  6463  unxpdomlem3  7069  dfac5lem4  7753  dcomex  8073  canthwelem  8272  uzrdgfni  11021  gsum2d2  15225  cbcpcp  25162
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-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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