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Theorem opeqsn 4278
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 3811 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43eqeq1i 2303 . 2  |-  ( <. A ,  B >.  =  { C }  <->  { { A } ,  { A ,  B } }  =  { C } )
5 snex 4232 . . 3  |-  { A }  e.  _V
6 prex 4233 . . 3  |-  { A ,  B }  e.  _V
7 opeqsn.3 . . 3  |-  C  e. 
_V
85, 6, 7preqsn 3808 . 2  |-  ( { { A } ,  { A ,  B } }  =  { C } 
<->  ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C
) )
9 eqcom 2298 . . . . 5  |-  ( { A }  =  { A ,  B }  <->  { A ,  B }  =  { A } )
101, 2, 1preqsn 3808 . . . . 5  |-  ( { A ,  B }  =  { A }  <->  ( A  =  B  /\  B  =  A ) )
11 eqcom 2298 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
1211anbi2i 675 . . . . . 6  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
13 anidm 625 . . . . . 6  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
1412, 13bitri 240 . . . . 5  |-  ( ( A  =  B  /\  B  =  A )  <->  A  =  B )
159, 10, 143bitri 262 . . . 4  |-  ( { A }  =  { A ,  B }  <->  A  =  B )
1615anbi1i 676 . . 3  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  { A ,  B }  =  C ) )
17 dfsn2 3667 . . . . . . 7  |-  { A }  =  { A ,  A }
18 preq2 3720 . . . . . . 7  |-  ( A  =  B  ->  { A ,  A }  =  { A ,  B }
)
1917, 18syl5req 2341 . . . . . 6  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2019eqeq1d 2304 . . . . 5  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  { A }  =  C ) )
21 eqcom 2298 . . . . 5  |-  ( { A }  =  C  <-> 
C  =  { A } )
2220, 21syl6bb 252 . . . 4  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  C  =  { A } ) )
2322pm5.32i 618 . . 3  |-  ( ( A  =  B  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
2416, 23bitri 240 . 2  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
254, 8, 243bitri 262 1  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   {cpr 3654   <.cop 3656
This theorem is referenced by:  relop  4850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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