Theorem zeneo 11568

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 9152 follows immediately from the fact that a contradiction implies anything, see pm2.21i 635. (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 3947	. 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 1480   =/= wne 2308   class class class wbr 3929   2c2 8771   ZZcz 9054   || cdvds 11493

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 603  ax-in2 604  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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

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-ne 2309  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930

This theorem is referenced by: (None)