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Mirrors > Home > MPE Home > Th. List > Mathboxes > gbeeven | Structured version Visualization version GIF version |
Description: An even Goldbach number is even. (Contributed by AV, 25-Jul-2020.) |
Ref | Expression |
---|---|
gbeeven | ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgbe 40173 | . 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞)))) | |
2 | 1 | simplbi 475 | 1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 (class class class)co 6549 + caddc 9818 ℙcprime 15223 Even ceven 40075 Odd codd 40076 GoldbachEven cgbe 40167 |
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-3an 1033 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-rex 2902 df-rab 2905 df-v 3175 df-gbe 40170 |
This theorem is referenced by: (None) |
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