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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 3092 | . 2 ⊢ (𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶) | |
2 | nelne1 3083 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐵 ≠ 𝐶) | |
3 | 1, 2 | sylan2b 596 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∉ 𝐶) → 𝐵 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ≠ wne 2987 ∉ wnel 3091 |
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-cleq 2791 df-clel 2870 df-ne 2988 df-nel 3092 |
This theorem is referenced by: (None) |
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