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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 3976 | . . . 4 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | |
2 | 1 | biimpi 208 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
3 | 2 | orcomd 860 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐶 ∨ 𝐴 ∈ 𝐵)) |
4 | 3 | orcanai 988 | 1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 ∨ wo 836 ∈ wcel 2107 ∪ cun 3790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-v 3400 df-un 3797 |
This theorem is referenced by: limcresiooub 40782 limcresioolb 40783 fourierdlem48 41298 fourierdlem49 41299 fourierdlem101 41351 prsal 41462 isomenndlem 41671 hsphoidmvle2 41726 hsphoidmvle 41727 |
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