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Theorem dfopif 4210
Description: Rewrite df-op 4034 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 4034 . 2  |-  <. A ,  B >.  =  { x  |  ( A  e. 
_V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } ) }
2 df-3an 975 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  x  e.  { { A } ,  { A ,  B } } )  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) )
32abbii 2601 . 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 3945 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  { { A } ,  { A ,  B } } )
5 ibar 504 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( x  e.  { { A } ,  { A ,  B } } 
<->  ( ( A  e. 
_V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) ) )
65abbi2dv 2604 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =  {
x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) } )
74, 6eqtr2d 2509 . . 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 468 . . . . . 6  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } )  ->  x  e.  (/) ) )
109abssdv 3574 . . . . 5  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  C_  (/) )
11 ss0 3816 . . . . 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 16 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  { x  |  ( ( A  e.  _V  /\  B  e.  _V )  /\  x  e.  { { A } ,  { A ,  B } } ) }  =  (/) )
13 iffalse 3948 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )  =  (/) )
1412, 13eqtr4d 2511 . . 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 2500 1  |-  <. A ,  B >.  =  if ( ( A  e.  _V  /\  B  e.  _V ) ,  { { A } ,  { A ,  B } } ,  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   {cab 2452   _Vcvv 3113    C_ wss 3476   (/)c0 3785   ifcif 3939   {csn 4027   {cpr 4029   <.cop 4033
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
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-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-op 4034
This theorem is referenced by:  dfopg  4211  opeq1  4213  opeq2  4214  nfop  4229  opprc  4235  opex  4711
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