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Theorem zeneo 11602
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 9175 follows immediately from the fact that a contradiction implies anything, see pm2.21i 636. (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 3954 . 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 103  wcel 1481  wne 2309   class class class wbr 3936  2c2 8794  cz 9077  cdvds 11527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937
This theorem is referenced by: (None)
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