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 5211. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
nvelim | ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvel 5211 | . 2 ⊢ ¬ V ∈ 𝐵 | |
2 | eleq1 2897 | . . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 2 | eqcoms 2826 | . 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
4 | 1, 3 | mtbii 327 | 1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-v 3494 |
This theorem is referenced by: afvvdm 43217 afvvfunressn 43219 afvvv 43221 afvvfveq 43224 |
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