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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 4360 | . 2 ⊢ ¬ 𝐵 ∈ ∅ | |
2 | eleq2 2833 | . 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 1537 ∈ wcel 2108 ∅c0 4352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-dif 3979 df-nul 4353 |
This theorem is referenced by: iresn0n0 6083 0mpo0 7533 nbgr0vtx 29390 disjxun0 32596 disjlem14 38754 oe0rif 43247 clnbgr0vtx 47708 |
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