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Theorem zeneo 14843
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 11288 follows immediately from the fact that a contradiction implies anything, see pm2.21i 114. (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 breq2 4577 . . . . 5 (𝐴 = 𝐵 → (2 ∥ 𝐴 ↔ 2 ∥ 𝐵))
21anbi1d 736 . . . 4 (𝐴 = 𝐵 → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) ↔ (2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵)))
3 pm3.24 921 . . . . 5 ¬ (2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵)
43pm2.21i 114 . . . 4 ((2 ∥ 𝐵 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵)
52, 4syl6bi 241 . . 3 (𝐴 = 𝐵 → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵))
65a1d 25 . 2 (𝐴 = 𝐵 → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵)))
7 neqne 2785 . . 3 𝐴 = 𝐵𝐴𝐵)
872a1d 26 . 2 𝐴 = 𝐵 → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵)))
96, 8pm2.61i 174 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  2c2 10913  cz 11206  cdvds 14763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574
This theorem is referenced by: (None)
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