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Theorem evenelz 14765
 Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 14693. (Contributed by AV, 22-Jun-2021.)
Assertion
Ref Expression
evenelz (2 ∥ 𝑁𝑁 ∈ ℤ)

Proof of Theorem evenelz
StepHypRef Expression
1 dvdszrcl 14693 . 2 (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
21simprd 477 1 (2 ∥ 𝑁𝑁 ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1938   class class class wbr 4481  2c2 10824  ℤcz 11117   ∥ cdvds 14688 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732 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 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-br 4482  df-opab 4542  df-xp 4938  df-dvds 14689 This theorem is referenced by:  even2n  14771
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