ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opth1 Unicode version

Theorem opth1 3973
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 3531 . . 3  |-  ( { A }  =  { C }  ->  A  =  C )
32a1i 9 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C }  ->  A  =  C ) )
4 opth1.2 . . . . . . . . 9  |-  B  e. 
_V
51, 4opi1 3969 . . . . . . . 8  |-  { A }  e.  <. A ,  B >.
6 id 19 . . . . . . . 8  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. A ,  B >.  = 
<. C ,  D >. )
75, 6syl5eleq 2126 . . . . . . 7  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  <. C ,  D >. )
8 oprcl 3573 . . . . . . 7  |-  ( { A }  e.  <. C ,  D >.  ->  ( C  e.  _V  /\  D  e.  _V ) )
97, 8syl 14 . . . . . 6  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( C  e.  _V  /\  D  e.  _V )
)
109simpld 105 . . . . 5  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  _V )
11 prid1g 3474 . . . . 5  |-  ( C  e.  _V  ->  C  e.  { C ,  D } )
1210, 11syl 14 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  C  e.  { C ,  D } )
13 eleq2 2101 . . . 4  |-  ( { A }  =  { C ,  D }  ->  ( C  e.  { A }  <->  C  e.  { C ,  D } ) )
1412, 13syl5ibrcom 146 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C ,  D }  ->  C  e.  { A } ) )
15 elsni 3393 . . . 4  |-  ( C  e.  { A }  ->  C  =  A )
1615eqcomd 2045 . . 3  |-  ( C  e.  { A }  ->  A  =  C )
1714, 16syl6 29 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C ,  D }  ->  A  =  C ) )
18 dfopg 3547 . . . . 5  |-  ( ( C  e.  _V  /\  D  e.  _V )  -> 
<. C ,  D >.  =  { { C } ,  { C ,  D } } )
197, 8, 183syl 17 . . . 4  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  <. C ,  D >.  =  { { C } ,  { C ,  D } } )
207, 19eleqtrd 2116 . . 3  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  { A }  e.  { { C } ,  { C ,  D } } )
21 elpri 3398 . . 3  |-  ( { A }  e.  { { C } ,  { C ,  D } }  ->  ( { A }  =  { C }  \/  { A }  =  { C ,  D } ) )
2220, 21syl 14 . 2  |-  ( <. A ,  B >.  = 
<. C ,  D >.  -> 
( { A }  =  { C }  \/  { A }  =  { C ,  D }
) )
233, 17, 22mpjaod 638 1  |-  ( <. A ,  B >.  = 
<. C ,  D >.  ->  A  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    \/ wo 629    = wceq 1243    e. wcel 1393   _Vcvv 2557   {csn 3375   {cpr 3376   <.cop 3378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384
This theorem is referenced by:  opth  3974  dmsnopg  4792  funcnvsn  4945  oprabid  5537
  Copyright terms: Public domain W3C validator