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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 16206. (Contributed by AV, 22-Jun-2021.) |
Ref | Expression |
---|---|
evenelz | ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdszrcl 16206 | . 2 ⊢ (2 ∥ 𝑁 → (2 ∈ ℤ ∧ 𝑁 ∈ ℤ)) | |
2 | 1 | simprd 494 | 1 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2104 class class class wbr 5147 2c2 12271 ℤcz 12562 ∥ cdvds 16201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-dvds 16202 |
This theorem is referenced by: even2n 16289 |
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