Theorem zeneo 11604

Description: No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. This variant of zneo 9176 follows immediately from the fact that a contradiction implies anything, see pm2.21i 636. (Contributed by AV, 22-Jun-2021.)

Assertion

Ref	Expression
zeneo	|- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) )

Proof of Theorem zeneo

Step	Hyp	Ref	Expression
1		nbrne1 3955	. 2 |- ( ( 2 || A /\ -. 2 || B ) -> A =/= B )
2	1	a1i 9	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 1481   =/= wne 2309   class class class wbr 3937   2c2 8795   ZZcz 9078   || cdvds 11529

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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938

This theorem is referenced by: (None)