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Theorem zeneo 10478
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 8581 follows immediately from the fact that a contradiction implies anything, see pm2.21i 608. (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
StepHypRef Expression
1 nbrne1 3822 . 2  |-  ( ( 2  ||  A  /\  -.  2  ||  B )  ->  A  =/=  B
)
21a1i 9 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( 2  ||  A  /\  -.  2  ||  B )  ->  A  =/=  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    e. wcel 1434    =/= wne 2249   class class class wbr 3805   2c2 8208   ZZcz 8484    || cdvds 10403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806
This theorem is referenced by: (None)
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