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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 41639 | . 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞)))) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 (class class class)co 6650 + caddc 9939 ℙcprime 15385 Even ceven 41537 Odd codd 41538 GoldbachEven cgbe 41633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-gbe 41636 |
This theorem is referenced by: (None) |
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