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Theorem difrab0eqim 3349
Description: If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
difrab0eqim  |-  ( V  =  { x  e.  V  |  ph }  ->  ( V  \  {
x  e.  V  |  ph } )  =  (/) )
Distinct variable group:    x, V
Allowed substitution hint:    ph( x)

Proof of Theorem difrab0eqim
StepHypRef Expression
1 ssrabeq 3107 . 2  |-  ( V 
C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph }
)
2 ssdif0im 3347 . 2  |-  ( V 
C_  { x  e.  V  |  ph }  ->  ( V  \  {
x  e.  V  |  ph } )  =  (/) )
31, 2sylbir 133 1  |-  ( V  =  { x  e.  V  |  ph }  ->  ( V  \  {
x  e.  V  |  ph } )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   {crab 2363    \ cdif 2996    C_ wss 2999   (/)c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by: (None)
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