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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 4149 | . 2 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | eleq2 2896 | . 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅)) | |
3 | 1, 2 | mtbiri 319 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1658 ∈ wcel 2166 ∅c0 4145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-v 3417 df-dif 3802 df-nul 4146 |
This theorem is referenced by: (None) |
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