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Theorem prcssprc 5197
 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 5195 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
21ex 416 . . 3 (𝐴𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
32nelcon3d 3103 . 2 (𝐴𝐵 → (𝐴 ∉ V → 𝐵 ∉ V))
43imp 410 1 ((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111   ∉ wnel 3091  Vcvv 3442   ⊆ wss 3883 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770  ax-sep 5171 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nel 3092  df-rab 3115  df-v 3444  df-in 3890  df-ss 3900 This theorem is referenced by:  usgrprc  27100  rgrusgrprc  27423  rgrprc  27425
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