Theorem zeneo 12182

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 9474 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 4063	. 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 2176   ≠ wne 2376   class class class wbr 4044  2c2 9087  ℤcz 9372   ∥ cdvds 12098

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045

This theorem is referenced by: (None)