Theorem zeneo 12382

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 9548 follows immediately from the fact that a contradiction implies anything, see pm2.21i 649. (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 4102	. 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 104   e. wcel 2200   =/= wne 2400   class class class wbr 4083   2c2 9161   ZZcz 9446   || cdvds 12298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084

This theorem is referenced by: (None)