![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3428 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | 1 | con3i 152 | 1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2166 Vcvv 3413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-12 2222 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-tru 1662 df-ex 1881 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-v 3415 |
This theorem is referenced by: fundmge2nop0 13562 fun2dmnop0 13564 vtxval 26297 iedgval 26298 eliin2f 40101 dfatprc 42031 afvprc 42045 afv2prc 42127 |
Copyright terms: Public domain | W3C validator |