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Mirrors > Home > MPE Home > Th. List > zeneo | Structured version Visualization version GIF version |
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 12089 follows immediately from the fact that a contradiction implies anything, see pm2.21i 119. (Contributed by AV, 22-Jun-2021.) |
Ref | Expression |
---|---|
zeneo | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nbrne1 5044 | . 2 ⊢ ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵) | |
2 | 1 | a1i 11 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ ¬ 2 ∥ 𝐵) → 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∈ wcel 2112 ≠ wne 2949 class class class wbr 5025 2c2 11714 ℤcz 12005 ∥ cdvds 15640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-ne 2950 df-v 3409 df-un 3859 df-sn 4516 df-pr 4518 df-op 4522 df-br 5026 |
This theorem is referenced by: (None) |
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