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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 3501 | . 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 2108 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 |
| This theorem is referenced by: suppco 8231 fundmge2nop0 14541 fun2dmnop0 14543 vtxval 29017 iedgval 29018 fmlafvel 35390 isinf2 37406 eliin2f 45109 dfatprc 47142 afvprc 47156 afv2prc 47238 |
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