Theorem evenelz 12378

Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 12303. (Contributed by AV, 22-Jun-2021.)

Assertion

Ref	Expression
evenelz	|- ( 2 || N -> N e. ZZ )

Proof of Theorem evenelz

Step	Hyp	Ref	Expression
1		dvdszrcl 12303	. 2 |- ( 2 || N -> ( 2 e. ZZ /\ N e. ZZ ) )
2	1	simprd 114	1 |- ( 2 || N -> N e. ZZ )

Syntax hints:   -> wi 4   e. wcel 2200   class class class wbr 4083   2c2 9161   ZZcz 9446   || cdvds 12298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-dvds 12299

This theorem is referenced by:  even2n  12385