Theorem zeneo 16048

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. This variant of zneo 12403 follows immediately from the fact that a contradiction implies anything, see pm2.21i 119. (Contributed by AV, 22-Jun-2021.)

Assertion

Ref	Expression
zeneo	⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))

Proof of Theorem zeneo

Step	Hyp	Ref	Expression
1		nbrne1 5093	. 2 ⊢ ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)
2	1	a1i 11	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2106   ≠ wne 2943   class class class wbr 5074  2c2 12028  ℤcz 12319   ∥ cdvds 15963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075

This theorem is referenced by: (None)