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Mirrors > Home > MPE Home > Th. List > zeo4 | Structured version Visualization version GIF version |
Description: An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 12063 follows immediately from the principle of double negation, see notnotb 317. (Contributed by AV, 17-Jun-2021.) |
Ref | Expression |
---|---|
zeo4 | ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnotb 317 | . 2 ⊢ (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁) | |
2 | 1 | a1i 11 | 1 ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∈ wcel 2110 class class class wbr 5058 2c2 11686 ℤcz 11975 ∥ cdvds 15601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 209 |
This theorem is referenced by: (None) |
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