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Mirrors > Home > MPE Home > Th. List > nel02 | Structured version Visualization version GIF version |
Description: The empty set has no elements. (Contributed by Peter Mazsa, 4-Jan-2018.) |
Ref | Expression |
---|---|
nel02 | ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4330 | . 2 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | eleq2 2821 | . 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅)) | |
3 | 1, 2 | mtbiri 327 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2105 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-dif 3951 df-nul 4323 |
This theorem is referenced by: iresn0n0 6053 0mpo0 7495 disjxun0 32237 disjlem14 38131 oe0rif 42497 |
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