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Mathbox for Alexander van der Vekens |
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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 45913 | . 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞)))) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3072 (class class class)co 7354 + caddc 11051 ℙcprime 16544 Even ceven 45786 Odd codd 45787 GoldbachEven cgbe 45907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rex 3073 df-rab 3407 df-v 3446 df-gbe 45910 |
This theorem is referenced by: (None) |
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