![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4247 | . 2 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | eleq2 2878 | . 2 ⊢ (𝐴 = ∅ → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ ∅)) | |
3 | 1, 2 | mtbiri 330 | 1 ⊢ (𝐴 = ∅ → ¬ 𝐵 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-dif 3884 df-nul 4244 |
This theorem is referenced by: iresn0n0 5890 0mpo0 7216 disjxun0 30337 |
Copyright terms: Public domain | W3C validator |