Theorem evenelz 15268

Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 15194. (Contributed by AV, 22-Jun-2021.)

Assertion

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

Proof of Theorem evenelz

Step	Hyp	Ref	Expression
1		dvdszrcl 15194	. 2 ⊢ (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2	1	simprd 483	1 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ)

Syntax hints:   → wi 4   ∈ wcel 2145   class class class wbr 4786  2c2 11272  ℤcz 11579   ∥ cdvds 15189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-dvds 15190

This theorem is referenced by:  even2n  15274