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Theorem evenelz 16283
Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16206. (Contributed by AV, 22-Jun-2021.)
Assertion
Ref Expression
evenelz (2 ∥ 𝑁𝑁 ∈ ℤ)

Proof of Theorem evenelz
StepHypRef Expression
1 dvdszrcl 16206 . 2 (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
21simprd 494 1 (2 ∥ 𝑁𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104   class class class wbr 5147  2c2 12271  cz 12562  cdvds 16201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-dvds 16202
This theorem is referenced by:  even2n  16289
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