Theorem evenelz 16393

Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16314. (Contributed by AV, 22-Jun-2021.)

Assertion

Ref	Expression
evenelz	⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ)

Proof of Theorem evenelz

Step	Hyp	Ref	Expression
1		dvdszrcl 16314	. 2 ⊢ (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2	1	simprd 500	1 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2149   class class class wbr 5113  2c2 12294  ℤcz 12590   ∥ cdvds 16309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-dvds 16310

This theorem is referenced by:  even2n  16399