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Theorem csbprc 3376
Description: The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
csbprc  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )

Proof of Theorem csbprc
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-csb 2974 . 2  |-  [_ A  /  x ]_ B  =  { y  |  [. A  /  x ]. y  e.  B }
2 sbcex 2888 . . . . . . 7  |-  ( [. A  /  x ]. y  e.  B  ->  A  e. 
_V )
32con3i 604 . . . . . 6  |-  ( -.  A  e.  _V  ->  -. 
[. A  /  x ]. y  e.  B
)
43pm2.21d 591 . . . . 5  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. y  e.  B  -> F.  ) )
5 falim 1328 . . . . 5  |-  ( F. 
->  [. A  /  x ]. y  e.  B
)
64, 5impbid1 141 . . . 4  |-  ( -.  A  e.  _V  ->  (
[. A  /  x ]. y  e.  B  <-> F.  ) )
76abbidv 2233 . . 3  |-  ( -.  A  e.  _V  ->  { y  |  [. A  /  x ]. y  e.  B }  =  {
y  | F.  }
)
8 fal 1321 . . . 4  |-  -. F.
98abf 3374 . . 3  |-  { y  | F.  }  =  (/)
107, 9syl6eq 2164 . 2  |-  ( -.  A  e.  _V  ->  { y  |  [. A  /  x ]. y  e.  B }  =  (/) )
111, 10syl5eq 2160 1  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1314   F. wfal 1319    e. wcel 1463   {cab 2101   _Vcvv 2658   [.wsbc 2880   [_csb 2973   (/)c0 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by: (None)
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