Theorem zeo4 16249

Description: An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 12563 follows immediately from the principle of double negation, see notnotb 315. (Contributed by AV, 17-Jun-2021.)

Assertion

Ref	Expression
zeo4	⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

Proof of Theorem zeo4

Step	Hyp	Ref	Expression
1		notnotb 315	. 2 ⊢ (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁)
2	1	a1i 11	1 ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∈ wcel 2109   class class class wbr 5092  2c2 12183  ℤcz 12471   ∥ cdvds 16163

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 207

This theorem is referenced by: (None)