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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 4147 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) | |
2 | 1 | biimpi 215 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
3 | 2 | orcanai 999 | 1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 843 ∈ wcel 2104 ∪ cun 3945 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-un 3952 |
This theorem is referenced by: fsumsplitsn 15694 fprodsplitsn 15937 founiiun0 44187 infxrpnf 44454 cnrefiisplem 44843 dvnprodlem1 44960 fourierdlem70 45190 fourierdlem71 45191 fourierdlem80 45200 sge0splitmpt 45425 sge0iunmptlemfi 45427 nnfoctbdjlem 45469 hoidmvlelem2 45610 hoidmvlelem3 45611 pimrecltpos 45722 |
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