Theorem zeneo 14901

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 11336 follows immediately from the fact that a contradiction implies anything, see pm2.21i 115. (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		breq2 4587	. . . . 5 ⊢ (𝐴 = 𝐵 → (2 ∥ 𝐴 ↔ 2 ∥ 𝐵))
2	1	anbi1d 737	. . . 4 ⊢ (𝐴 = 𝐵 → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) ↔ (2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵)))
3		pm3.24 922	. . . . 5 ⊢ ¬ (2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵)
4	3	pm2.21i 115	. . . 4 ⊢ ((2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)
5	2, 4	syl6bi 242	. . 3 ⊢ (𝐴 = 𝐵 → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))
6	5	a1d 25	. 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)))
7		neqne 2790	. . 3 ⊢ (¬ 𝐴 = 𝐵 → 𝐴 ≠ 𝐵)
8	7	2a1d 26	. 2 ⊢ (¬ 𝐴 = 𝐵 → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)))
9	6, 8	pm2.61i 175	1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  2c2 10947  ℤcz 11254   ∥ cdvds 14821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584

This theorem is referenced by: (None)