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Theorem difrab0eqim 3481
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 3234 . 2  |-  ( V 
C_  { x  e.  V  |  ph }  <->  V  =  { x  e.  V  |  ph }
)
2 ssdif0im 3479 . 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 1348   {crab 2452    \ cdif 3118    C_ wss 3121   (/)c0 3414
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415
This theorem is referenced by: (None)
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