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Mirrors > Home > MPE Home > Th. List > Mathboxes > elunnel2 | Structured version Visualization version GIF version |
Description: A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
elunnel2 | ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3715 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | |
2 | 1 | biimpi 205 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
3 | 2 | orcomd 402 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐶 ∨ 𝐴 ∈ 𝐵)) |
4 | 3 | orcanai 950 | 1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 ∈ wcel 1977 ∪ cun 3538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-un 3545 |
This theorem is referenced by: limcresiooub 38709 limcresioolb 38710 fourierdlem48 39047 fourierdlem49 39048 fourierdlem101 39100 prsal 39214 isomenndlem 39420 hsphoidmvle2 39475 hsphoidmvle 39476 |
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