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Mirrors > Home > MPE Home > Th. List > prcnel | Structured version Visualization version GIF version |
Description: A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.) |
Ref | Expression |
---|---|
prcnel | ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | 1 | con3i 149 | 1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-tru 1478 df-ex 1696 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: nelsnOLD 4160 fundmge2nop0 13129 fun2dmnop0 13131 eliin2f 38316 |
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