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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 5077. (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
nvelim | ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvel 5077 | . 2 ⊢ ¬ V ∈ 𝐵 | |
2 | eleq1 2854 | . . 3 ⊢ (V = 𝐴 → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
3 | 2 | eqcoms 2787 | . 2 ⊢ (𝐴 = V → (V ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
4 | 1, 3 | mtbii 318 | 1 ⊢ (𝐴 = V → ¬ 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 Vcvv 3416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-ext 2751 ax-sep 5060 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-v 3418 |
This theorem is referenced by: afvvdm 42744 afvvfunressn 42746 afvvv 42748 afvvfveq 42751 |
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