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Theorem opeqsn 4697
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 4157 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43eqeq1i 2476 . 2  |-  ( <. A ,  B >.  =  { C }  <->  { { A } ,  { A ,  B } }  =  { C } )
5 snex 4641 . . 3  |-  { A }  e.  _V
6 prex 4642 . . 3  |-  { A ,  B }  e.  _V
7 opeqsn.3 . . 3  |-  C  e. 
_V
85, 6, 7preqsn 4151 . 2  |-  ( { { A } ,  { A ,  B } }  =  { C } 
<->  ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C
) )
9 eqcom 2478 . . . . 5  |-  ( { A }  =  { A ,  B }  <->  { A ,  B }  =  { A } )
101, 2, 1preqsn 4151 . . . . 5  |-  ( { A ,  B }  =  { A }  <->  ( A  =  B  /\  B  =  A ) )
11 eqcom 2478 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
1211anbi2i 708 . . . . . 6  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
13 anidm 656 . . . . . 6  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
1412, 13bitri 257 . . . . 5  |-  ( ( A  =  B  /\  B  =  A )  <->  A  =  B )
159, 10, 143bitri 279 . . . 4  |-  ( { A }  =  { A ,  B }  <->  A  =  B )
1615anbi1i 709 . . 3  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  { A ,  B }  =  C ) )
17 dfsn2 3972 . . . . . . 7  |-  { A }  =  { A ,  A }
18 preq2 4043 . . . . . . 7  |-  ( A  =  B  ->  { A ,  A }  =  { A ,  B }
)
1917, 18syl5req 2518 . . . . . 6  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2019eqeq1d 2473 . . . . 5  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  { A }  =  C ) )
21 eqcom 2478 . . . . 5  |-  ( { A }  =  C  <-> 
C  =  { A } )
2220, 21syl6bb 269 . . . 4  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  C  =  { A } ) )
2322pm5.32i 649 . . 3  |-  ( ( A  =  B  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
2416, 23bitri 257 . 2  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
254, 8, 243bitri 279 1  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   {csn 3959   {cpr 3961   <.cop 3965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966
This theorem is referenced by:  relop  4990  snopeqop  39145  propeqop  39146
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