Theorem zneoALTV

Description: No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of Apostol p. 28. (Contributed by NM, 31-Jul-2004) (Revised by AV, 16-Jun-2020)

	Ref	Expression
Assertion	zneoALTV	⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		oddneven	⊢ ( 𝐵 ∈ Odd → ¬ 𝐵 ∈ Even )
2		nelne2	⊢ ( ( 𝐴 ∈ Even ∧ ¬ 𝐵 ∈ Even ) → 𝐴 ≠ 𝐵 )
3	1 2	sylan2	⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → 𝐴 ≠ 𝐵 )