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Theorem eluzelz2 42616
Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
eluzelz2.1 𝑍 = (ℤ𝑀)
Assertion
Ref Expression
eluzelz2 (𝑁𝑍𝑁 ∈ ℤ)

Proof of Theorem eluzelz2
StepHypRef Expression
1 eluzelz2.1 . . . 4 𝑍 = (ℤ𝑀)
21eleq2i 2829 . . 3 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
32biimpi 219 . 2 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
4 eluzelz 12448 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
53, 4syl 17 1 (𝑁𝑍𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  cfv 6380  cz 12176  cuz 12438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-cnex 10785  ax-resscn 10786
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-neg 11065  df-z 12177  df-uz 12439
This theorem is referenced by:  eluzelz2d  42626  uzublem  42643  uzinico  42773  limsupubuzlem  42928  limsupmnfuzlem  42942  limsupre3uzlem  42951  limsupvaluz2  42954  supcnvlimsup  42956  xlimclim2lem  43055  climxlim2  43062  xlimliminflimsup  43078  smflimmpt  44015  smflimsuplem3  44027  smflimsuplem4  44028  smflimsuplem5  44029  smflimsuplem6  44030  smflimsuplem7  44031  smflimsuplem8  44032  smflimsupmpt  44034  smfliminflem  44035  smfliminfmpt  44037
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