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Mirrors > Home > MPE Home > Th. List > disjel | Structured version Visualization version GIF version |
Description: A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.) |
Ref | Expression |
---|---|
disjel | ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj3 4220 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ 𝐴 = (𝐴 ∖ 𝐵)) | |
2 | eleq2 2871 | . . . 4 ⊢ (𝐴 = (𝐴 ∖ 𝐵) → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ (𝐴 ∖ 𝐵))) | |
3 | eldifn 3935 | . . . 4 ⊢ (𝐶 ∈ (𝐴 ∖ 𝐵) → ¬ 𝐶 ∈ 𝐵) | |
4 | 2, 3 | syl6bi 245 | . . 3 ⊢ (𝐴 = (𝐴 ∖ 𝐵) → (𝐶 ∈ 𝐴 → ¬ 𝐶 ∈ 𝐵)) |
5 | 1, 4 | sylbi 209 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐶 ∈ 𝐴 → ¬ 𝐶 ∈ 𝐵)) |
6 | 5 | imp 396 | 1 ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∖ cdif 3770 ∩ cin 3772 ∅c0 4119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2781 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2790 df-cleq 2796 df-clel 2799 df-nfc 2934 df-ral 3098 df-v 3391 df-dif 3776 df-in 3780 df-nul 4120 |
This theorem is referenced by: disjxun 4845 fvun1 6498 dedekindle 10495 fprodsplit 15037 unelldsys 30741 dvasin 33988 |
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