Theorem zeo4 15950

Description: An integer is even or odd but not both. With this representation of even and odd integers, this variant of zeo2 12312 follows immediately from the principle of double negation, see notnotb 318. (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 318	. 2 ⊢ (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁)
2	1	a1i 11	1 ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ¬ ¬ 2 ∥ 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∈ wcel 2112   class class class wbr 5070  2c2 11933  ℤcz 12224   ∥ cdvds 15866

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 210

This theorem is referenced by: (None)