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	$$⊢\left(A\in \mathrm{Even}\wedge B\in \mathrm{Odd}\right)\to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		oddneven	$$⊢B\in \mathrm{Odd}\to \neg B\in \mathrm{Even}$$
2		nelne2	$$⊢\left(A\in \mathrm{Even}\wedge \neg B\in \mathrm{Even}\right)\to A\ne B$$
3	1 2	sylan2	$$⊢\left(A\in \mathrm{Even}\wedge B\in \mathrm{Odd}\right)\to A\ne B$$