Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nvelim | Structured version Visualization version GIF version |
Description: If a class is the universal class it doesn't belong to any class, generalisation of nvel 4725. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
nvelim | ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvel 4725 | . 2 ⊢ ¬ V ∈ 𝐵 | |
2 | eleq1 2676 | . . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 2 | eqcoms 2618 | . 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
4 | 1, 3 | mtbii 315 | 1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ 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-8 1979 ax-9 1986 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
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: afvvdm 39870 afvvfunressn 39872 afvvv 39874 afvvfveq 39877 |
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