Theorem evenelz 12578

Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 12503. (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 12503	. 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 2205   class class class wbr 4114   2c2 9305   ZZcz 9594   || cdvds 12498

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-dvds 12499

This theorem is referenced by:  even2n  12585