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Mirrors > Home > MPE Home > Th. List > elnelne1 | Structured version Visualization version GIF version |
Description: Two classes are different if they don't contain the same element. (Contributed by AV, 28-Jan-2020.) |
Ref | Expression |
---|---|
elnelne1 | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nel 3045 | . 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶) | |
2 | nelne1 3037 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶) | |
3 | 1, 2 | sylan2b 594 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ∉ wnel 3044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-clel 2814 df-ne 2939 df-nel 3045 |
This theorem is referenced by: sticksstones1 42128 |
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