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Theorem difrab0eqim 3399
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 3153 . 2  |-  ( V 
C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph }
)
2 ssdif0im 3397 . 2  |-  ( V 
C_  { x  e.  V  |  ph }  ->  ( V  \  {
x  e.  V  |  ph } )  =  (/) )
31, 2sylbir 134 1  |-  ( V  =  { x  e.  V  |  ph }  ->  ( V  \  {
x  e.  V  |  ph } )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   {crab 2397    \ cdif 3038    C_ wss 3041   (/)c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334
This theorem is referenced by: (None)
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