| 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 5316. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| Ref | Expression |
|---|---|
| nvelim | ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nvel 5316 | . 2 ⊢ ¬ V ∈ 𝐵 | |
| 2 | eleq1 2829 | . . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
| 3 | 2 | eqcoms 2745 | . 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 1540 ∈ 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 ax-sep 5296 |
| 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: afvvdm 47153 afvvfunressn 47155 afvvv 47157 afvvfveq 47160 |
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