Theorem prcssprc 5231

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 5229	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2	1	ex 415	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ V → 𝐴 ∈ V))
3	2	nelcon3d 3137	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∉ V → 𝐵 ∉ V))
4	3	imp 409	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114   ∉ wnel 3125  Vcvv 3496   ⊆ wss 3938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-rab 3149  df-v 3498  df-in 3945  df-ss 3954

This theorem is referenced by:  usgrprc  27050  rgrusgrprc  27373  rgrprc  27375