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Theorem opeqsn 4749
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 4218 . . 3  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
43eqeq1i 2474 . 2  |-  ( <. A ,  B >.  =  { C }  <->  { { A } ,  { A ,  B } }  =  { C } )
5 snex 4694 . . 3  |-  { A }  e.  _V
6 prex 4695 . . 3  |-  { A ,  B }  e.  _V
7 opeqsn.3 . . 3  |-  C  e. 
_V
85, 6, 7preqsn 4215 . 2  |-  ( { { A } ,  { A ,  B } }  =  { C } 
<->  ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C
) )
9 eqcom 2476 . . . . 5  |-  ( { A }  =  { A ,  B }  <->  { A ,  B }  =  { A } )
101, 2, 1preqsn 4215 . . . . 5  |-  ( { A ,  B }  =  { A }  <->  ( A  =  B  /\  B  =  A ) )
11 eqcom 2476 . . . . . . 7  |-  ( B  =  A  <->  A  =  B )
1211anbi2i 694 . . . . . 6  |-  ( ( A  =  B  /\  B  =  A )  <->  ( A  =  B  /\  A  =  B )
)
13 anidm 644 . . . . . 6  |-  ( ( A  =  B  /\  A  =  B )  <->  A  =  B )
1412, 13bitri 249 . . . . 5  |-  ( ( A  =  B  /\  B  =  A )  <->  A  =  B )
159, 10, 143bitri 271 . . . 4  |-  ( { A }  =  { A ,  B }  <->  A  =  B )
1615anbi1i 695 . . 3  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  { A ,  B }  =  C ) )
17 dfsn2 4046 . . . . . . 7  |-  { A }  =  { A ,  A }
18 preq2 4113 . . . . . . 7  |-  ( A  =  B  ->  { A ,  A }  =  { A ,  B }
)
1917, 18syl5req 2521 . . . . . 6  |-  ( A  =  B  ->  { A ,  B }  =  { A } )
2019eqeq1d 2469 . . . . 5  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  { A }  =  C ) )
21 eqcom 2476 . . . . 5  |-  ( { A }  =  C  <-> 
C  =  { A } )
2220, 21syl6bb 261 . . . 4  |-  ( A  =  B  ->  ( { A ,  B }  =  C  <->  C  =  { A } ) )
2322pm5.32i 637 . . 3  |-  ( ( A  =  B  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
2416, 23bitri 249 . 2  |-  ( ( { A }  =  { A ,  B }  /\  { A ,  B }  =  C )  <->  ( A  =  B  /\  C  =  { A } ) )
254, 8, 243bitri 271 1  |-  ( <. A ,  B >.  =  { C }  <->  ( A  =  B  /\  C  =  { A } ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033   {cpr 4035   <.cop 4039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040
This theorem is referenced by:  relop  5159
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