Theorem evenz 41543

Description: An even number is an integer. (Contributed by AV, 14-Jun-2020.)

Assertion

Ref	Expression
evenz	|- ( Z e. Even -> Z e. ZZ )

Proof of Theorem evenz

Step	Hyp	Ref	Expression
1		iseven 41541	. 2 |- ( Z e. Even <-> ( Z e. ZZ /\ ( Z / 2 ) e. ZZ ) )
2	1	simplbi 476	1 |- ( Z e. Even -> Z e. ZZ )

Syntax hints:   -> wi 4   e. wcel 1990  (class class class)co 6650   / cdiv 10684   2c2 11070   ZZcz 11377   Even ceven 41537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-even 41539

This theorem is referenced by:  evenm1odd  41552  evenp1odd  41553  bits0eALTV  41591  opeoALTV  41595  omeoALTV  41597  epoo  41612  emoo  41613  epee  41614  emee  41615  evensumeven  41616  evenltle  41626  even3prm2  41628  mogoldbblem  41629  sbgoldbalt  41669  sgoldbeven3prm  41671  mogoldbb  41673  bgoldbachlt  41701  tgblthelfgott  41703  bgoldbachltOLD  41707  tgblthelfgottOLD  41709