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Theorem prcssprc 5280
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 5278 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
21ex 413 . . 3 (𝐴𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
32nelcon3d 3059 . 2 (𝐴𝐵 → (𝐴 ∉ V → 𝐵 ∉ V))
43imp 407 1 ((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wnel 3047  Vcvv 3443  wss 3908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nel 3048  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925
This theorem is referenced by:  usgrprc  28059  rgrusgrprc  28382  rgrprc  28384  fsetprcnexALT  45191
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