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Theorem opeqsn 4444
Description: Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
Hypotheses
Ref Expression
opeqsn.1  |-  A  e. 
_V
opeqsn.2  |-  B  e. 
_V
opeqsn.3  |-  C  e. 
_V
Assertion
Ref Expression
opeqsn  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )

Proof of Theorem opeqsn
StepHypRef Expression
1 opeqsn.1 . . . 4  |-  A  e. 
_V
2 opeqsn.2 . . . 4  |-  B  e. 
_V
31, 2dfop 3975 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43eqeq1i 2442 . 2  |-  ( <. A ,  B >.  =  { C }  <->  { { A } ,  { A ,  B } }  =  { C } )
5 snex 4397 . . 3  |-  { A }  e.  _V
6 prex 4398 . . 3  |-  { A ,  B }  e.  _V
7 opeqsn.3 . . 3  |-  C  e. 
_V
85, 6, 7preqsn 3972 . 2  |-  ( { { A } ,  { A ,  B } }  =  { C } 
<->  ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C
) )
9 eqcom 2437 . . . . 5  |-  ( { A }  =  { A ,  B }  <->  { A ,  B }  =  { A } )
101, 2, 1preqsn 3972 . . . . 5  |-  ( { A ,  B }  =  { A }  <->  ( A  =  B  /\  B  =  A ) )
11 eqcom 2437 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
1211anbi2i 676 . . . . . 6  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
13 anidm 626 . . . . . 6  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
1412, 13bitri 241 . . . . 5  |-  ( ( A  =  B  /\  B  =  A )  <->  A  =  B )
159, 10, 143bitri 263 . . . 4  |-  ( { A }  =  { A ,  B }  <->  A  =  B )
1615anbi1i 677 . . 3  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  { A ,  B }  =  C ) )
17 dfsn2 3820 . . . . . . 7  |-  { A }  =  { A ,  A }
18 preq2 3876 . . . . . . 7  |-  ( A  =  B  ->  { A ,  A }  =  { A ,  B }
)
1917, 18syl5req 2480 . . . . . 6  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2019eqeq1d 2443 . . . . 5  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  { A }  =  C ) )
21 eqcom 2437 . . . . 5  |-  ( { A }  =  C  <-> 
C  =  { A } )
2220, 21syl6bb 253 . . . 4  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  C  =  { A } ) )
2322pm5.32i 619 . . 3  |-  ( ( A  =  B  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
2416, 23bitri 241 . 2  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
254, 8, 243bitri 263 1  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   {cpr 3807   <.cop 3809
This theorem is referenced by:  relop  5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815
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