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Theorem vdif0im 3459
Description: Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
Assertion
Ref Expression
vdif0im  |-  ( A  =  _V  ->  ( _V  \  A )  =  (/) )

Proof of Theorem vdif0im
StepHypRef Expression
1 vss 3441 . 2  |-  ( _V  C_  A  <->  A  =  _V )
2 ssdif0im 3458 . 2  |-  ( _V  C_  A  ->  ( _V 
\  A )  =  (/) )
31, 2sylbir 134 1  |-  ( A  =  _V  ->  ( _V  \  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335   _Vcvv 2712    \ cdif 3099    C_ wss 3102   (/)c0 3394
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395
This theorem is referenced by: (None)
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