Theorem prcssprc 5244

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 5242	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2	1	ex 412	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
3	2	nelcon3d 3060	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∉ V → 𝐵 ∉ V))
4	3	imp 406	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ∉ wnel 3048  Vcvv 3422   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nel 3049  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by:  usgrprc  27536  rgrusgrprc  27859  rgrprc  27861  fsetprcnexALT  44443