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Mirrors > Home > MPE Home > Th. List > evenelz | Structured version Visualization version GIF version |
Description: An even number is an integer. This follows immediately from the reverse closure of the divides relation, see dvdszrcl 16076. (Contributed by AV, 22-Jun-2021.) |
Ref | Expression |
---|---|
evenelz | ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdszrcl 16076 | . 2 ⊢ (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
2 | 1 | simprd 497 | 1 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 class class class wbr 5104 2c2 12142 ℤcz 12433 ∥ cdvds 16071 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-xp 5637 df-dvds 16072 |
This theorem is referenced by: even2n 16159 |
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