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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 3485 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | 1 | con3i 154 | 1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2098 Vcvv 3466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 |
This theorem is referenced by: suppco 8186 fundmge2nop0 14449 fun2dmnop0 14451 vtxval 28695 iedgval 28696 fmlafvel 34831 isinf2 36742 eliin2f 44247 dfatprc 46289 afvprc 46303 afv2prc 46385 |
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