Theorem zeneo 15063

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 11460 follows immediately from the fact that a contradiction implies anything, see pm2.21i 116. (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 4672	. 2 |- ( ( 2 || A /\ -. 2 || B ) -> A =/= B )
2	1	a1i 11	1 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( 2 || A /\ -. 2 || B ) -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   class class class wbr 4653   2c2 11070   ZZcz 11377   || cdvds 14983

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-ne 2795  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-br 4654

This theorem is referenced by: (None)