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Theorem evenelz 11575
Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 11509. (Contributed by AV, 22-Jun-2021.)
Assertion
Ref Expression
evenelz (2 ∥ 𝑁𝑁 ∈ ℤ)

Proof of Theorem evenelz
StepHypRef Expression
1 dvdszrcl 11509 . 2 (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ))
21simprd 113 1 (2 ∥ 𝑁𝑁 ∈ ℤ)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480   class class class wbr 3929  2c2 8783  cz 9066  cdvds 11504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-dvds 11505
This theorem is referenced by:  even2n  11582
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