Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 44269 | . 2 ⊢ (𝑍 ∈ GoldbachEven ↔ (𝑍 ∈ Even ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑍 = (𝑝 + 𝑞)))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑍 ∈ GoldbachEven → 𝑍 ∈ Even ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 (class class class)co 7135 + caddc 10529 ℙcprime 16005 Even ceven 44142 Odd codd 44143 GoldbachEven cgbe 44263 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rex 3112 df-rab 3115 df-v 3443 df-gbe 44266 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |