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Theorem opth1 4560
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 4035 . . 3  |-  ( { A }  =  { C }  ->  A  =  C )
32a1i 11 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C }  ->  A  =  C ) )
4 opth1.2 . . . . . . . . 9  |-  B  e. 
_V
51, 4opi1 4554 . . . . . . . 8  |-  { A }  e.  <. A ,  B >.
6 id 22 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  D >. )
75, 6syl5eleq 2524 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  <. C ,  D >. )
8 oprcl 4079 . . . . . . 7  |-  ( { A }  e.  <. C ,  D >.  ->  ( C  e.  _V  /\  D  e.  _V ) )
97, 8syl 16 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( C  e.  _V  /\  D  e.  _V )
)
109simpld 459 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  _V )
11 prid1g 3976 . . . . 5  |-  ( C  e.  _V  ->  C  e.  { C ,  D } )
1210, 11syl 16 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  { C ,  D } )
13 eleq2 2499 . . . 4  |-  ( { A }  =  { C ,  D }  ->  ( C  e.  { A }  <->  C  e.  { C ,  D } ) )
1412, 13syl5ibrcom 222 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C ,  D }  ->  C  e.  { A } ) )
15 elsni 3897 . . . 4  |-  ( C  e.  { A }  ->  C  =  A )
1615eqcomd 2443 . . 3  |-  ( C  e.  { A }  ->  A  =  C )
1714, 16syl6 33 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C ,  D }  ->  A  =  C ) )
18 dfopg 4052 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  -> 
<. C ,  D >.  =  { { C } ,  { C ,  D } } )
197, 8, 183syl 20 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  D } } )
207, 19eleqtrd 2514 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  { { C } ,  { C ,  D } } )
21 elpri 3892 . . 3  |-  ( { A }  e.  { { C } ,  { C ,  D } }  ->  ( { A }  =  { C }  \/  { A }  =  { C ,  D } ) )
2220, 21syl 16 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C }  \/  { A }  =  { C ,  D }
) )
233, 17, 22mpjaod 381 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   {csn 3872   {cpr 3874   <.cop 3878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879
This theorem is referenced by:  opth  4561  dmsnopg  5305  funcnvsn  5458  oprabid  6110  seqomlem2  6898  unxpdomlem3  7511  dfac5lem4  8288  dcomex  8608  canthwelem  8809  uzrdgfni  11773  gsum2d2  16456  2trllemA  23417  2pthon  23469  2pthon3v  23471  constr3lem2  23500
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