Theorem prcssprc 5282

Description: The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)

Assertion

Ref	Expression
prcssprc	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)

Proof of Theorem prcssprc

Step	Hyp	Ref	Expression
1		ssexg 5278	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2	1	ex 416	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
3	2	nelcon3d 3064	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∉ V → 𝐵 ∉ V))
4	3	imp 410	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141   ∉ wnel 3060  Vcvv 3453   ⊆ wss 3904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nel 3061  df-rab 3414  df-v 3455  df-in 3911  df-ss 3921

This theorem is referenced by:  usgrprc  29413  rgrusgrprc  29736  rgrprc  29738  fsetprcnexALT  47620