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Theorem prcssprc 5283
Description: The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
Assertion
Ref Expression
prcssprc ((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)

Proof of Theorem prcssprc
StepHypRef Expression
1 ssexg 5281 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
21ex 414 . . 3 (𝐴𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
32nelcon3d 3058 . 2 (𝐴𝐵 → (𝐴 ∉ V → 𝐵 ∉ V))
43imp 408 1 ((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wnel 3046  Vcvv 3444  wss 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3047  df-rab 3407  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by:  usgrprc  28256  rgrusgrprc  28579  rgrprc  28581  fsetprcnexALT  45382
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