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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, generalization of nvel 5318. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
nvelim | ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvel 5318 | . 2 ⊢ ¬ V ∈ 𝐵 | |
2 | eleq1 2825 | . . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 2 | eqcoms 2741 | . 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
4 | 1, 3 | mtbii 326 | 1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1535 ∈ wcel 2104 Vcvv 3477 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 ax-sep 5301 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1538 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-v 3479 |
This theorem is referenced by: afvvdm 47048 afvvfunressn 47050 afvvv 47052 afvvfveq 47055 |
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