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Mirrors > Home > MPE Home > Th. List > prcssprc | Structured version Visualization version GIF version |
Description: The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
prcssprc | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 5278 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | ex 413 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ V → 𝐴 ∈ V)) |
3 | 2 | nelcon3d 3059 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∉ V → 𝐵 ∉ V)) |
4 | 3 | imp 407 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∉ wnel 3047 Vcvv 3443 ⊆ wss 3908 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nel 3048 df-rab 3406 df-v 3445 df-in 3915 df-ss 3925 |
This theorem is referenced by: usgrprc 28059 rgrusgrprc 28382 rgrprc 28384 fsetprcnexALT 45191 |
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