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Theorem prcssprc 4937
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 4935 . . . 4 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
21ex 397 . . 3 (𝐴𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
32nelcon3d 3057 . 2 (𝐴𝐵 → (𝐴 ∉ V → 𝐵 ∉ V))
43imp 393 1 ((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2144  wnel 3045  Vcvv 3349  wss 3721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-nel 3046  df-v 3351  df-in 3728  df-ss 3735
This theorem is referenced by:  usgrprc  26380  rgrusgrprc  26719  rgrprc  26721
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