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Theorem dfopif 4156
Description: Rewrite df-op 3979 using  if. When both arguments are sets, it reduces to the standard Kuratowski definition; otherwise, it is defined to be the empty set. Avoid directly depending on this detail so that theorems will not depend on the Kuratowski construction. (Contributed by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Assertion
Ref Expression
dfopif  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )

Proof of Theorem dfopif
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-op 3979 . 2  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
2 df-3an 976 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) )
32abbii 2536 . 2  |-  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  { x  |  (
( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }
4 iftrue 3891 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  { { A } ,  { A ,  B } } )
5 ibar 502 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( x  e.  { { A } ,  { A ,  B } } 
<->  ( ( A  e. 
_V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) ) )
65abbi2dv 2539 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =  {
x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) } )
74, 6eqtr2d 2444 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) ) )
8 pm2.21 108 . . . . . . 7  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ( A  e. 
_V  /\  B  e.  _V )  ->  x  e.  (/) ) )
98adantrd 466 . . . . . 6  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } )  ->  x  e.  (/) ) )
109abssdv 3513 . . . . 5  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  C_  (/) )
11 ss0 3770 . . . . 5  |-  ( { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  C_  (/)  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  (/) )
1210, 11syl 17 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  (/) )
13 iffalse 3894 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  (/) )
1412, 13eqtr4d 2446 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) ) )
157, 14pm2.61i 164 . 2  |-  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  if ( ( A  e. 
_V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
161, 3, 153eqtri 2435 1  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   {cab 2387   _Vcvv 3059    C_ wss 3414   (/)c0 3738   ifcif 3885   {csn 3972   {cpr 3974   <.cop 3978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-dif 3417  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-op 3979
This theorem is referenced by:  dfopg  4157  opeq1  4159  opeq2  4160  nfop  4175  opprc  4181  opex  4655
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