Theorem zeneo 12297

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 9509 follows immediately from the fact that a contradiction implies anything, see pm2.21i 647. (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 4078	. 2 ⊢ ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)
2	1	a1i 9	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2178   ≠ wne 2378   class class class wbr 4059  2c2 9122  ℤcz 9407   ∥ cdvds 12213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by: (None)