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Theorem prcssprc 5220
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 5218 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
21ex 413 . . 3 (𝐴𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
32nelcon3d 3132 . 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 2105  wnel 3120  Vcvv 3492  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-nel 3121  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949
This theorem is referenced by:  usgrprc  26975  rgrusgrprc  27298  rgrprc  27300
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