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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 member of the first class, is 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 4049 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | |
2 | 1 | biimpi 219 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
3 | 2 | orcanai 1003 | 1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 847 ∈ wcel 2112 ∪ cun 3851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-un 3858 |
This theorem is referenced by: fsumsplitsn 15272 fprodsplitsn 15514 founiiun0 42342 infxrpnf 42601 cnrefiisplem 42988 dvnprodlem1 43105 fourierdlem70 43335 fourierdlem71 43336 fourierdlem80 43345 sge0splitmpt 43567 sge0iunmptlemfi 43569 nnfoctbdjlem 43611 hoidmvlelem2 43752 hoidmvlelem3 43753 pimrecltpos 43861 |
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