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| Mirrors > Home > MPE Home > Th. List > elunnel1 | Structured version Visualization version GIF version | ||
| Description: A member of a union that is not a member of the first class, is a member of the second class. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| elunnel1 | ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elun 4153 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | |
| 2 | 1 | biimpi 216 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
| 3 | 2 | orcanai 1005 | 1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 ∈ wcel 2108 ∪ cun 3949 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 |
| This theorem is referenced by: fsumsplitsn 15780 fprodsplitsn 16025 founiiun0 45195 infxrpnf 45457 cnrefiisplem 45844 dvnprodlem1 45961 fourierdlem70 46191 fourierdlem71 46192 fourierdlem80 46201 sge0splitmpt 46426 sge0iunmptlemfi 46428 nnfoctbdjlem 46470 hoidmvlelem2 46611 hoidmvlelem3 46612 pimrecltpos 46723 |
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