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Theorem evenelz 16153
Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16076. (Contributed by AV, 22-Jun-2021.)
Assertion
Ref Expression
evenelz (2 ∥ 𝑁𝑁 ∈ ℤ)

Proof of Theorem evenelz
StepHypRef Expression
1 dvdszrcl 16076 . 2 (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
21simprd 497 1 (2 ∥ 𝑁𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5104  2c2 12142  cz 12433  cdvds 16071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5637  df-dvds 16072
This theorem is referenced by:  even2n  16159
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