Theorem evenelz 16274

Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16197. (Contributed by AV, 22-Jun-2021.)

Assertion

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

Proof of Theorem evenelz

Step	Hyp	Ref	Expression
1		dvdszrcl 16197	. 2 ⊢ (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2	1	simprd 497	1 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2107   class class class wbr 5146  2c2 12262  ℤcz 12553   ∥ cdvds 16192

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 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425

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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-br 5147  df-opab 5209  df-xp 5680  df-dvds 16193

This theorem is referenced by:  even2n  16280