Theorem evenelz 16429

Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16350. (Contributed by AV, 22-Jun-2021.)

Assertion

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

Proof of Theorem evenelz

Step	Hyp	Ref	Expression
1		dvdszrcl 16350	. 2 ⊢ (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
2	1	simprd 501	1 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   class class class wbr 5103  2c2 12322  ℤcz 12618   ∥ cdvds 16345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-dvds 16346

This theorem is used by:  even2n  16435