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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 4266 | . 2 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | eleq2 2827 | . 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 1539 ∈ wcel 2106 ∅c0 4258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-dif 3891 df-nul 4259 |
This theorem is referenced by: iresn0n0 5965 0mpo0 7358 disjxun0 30910 |
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