ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  zeneo GIF version

Theorem zeneo 10404
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 8518 follows immediately from the fact that a contradiction implies anything, see pm2.21i 608. (Contributed by AV, 22-Jun-2021.)
Assertion
Ref Expression
zeneo ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵))

Proof of Theorem zeneo
StepHypRef Expression
1 nbrne1 3804 . 2 ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵)
21a1i 9 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wcel 1434  wne 2246   class class class wbr 3787  2c2 8145  cz 8421  cdvds 10329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-v 2604  df-un 2978  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator