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Theorem eluzelz2 44099
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 2825 . . 3 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
32biimpi 215 . 2 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
4 eluzelz 12828 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
53, 4syl 17 1 (𝑁𝑍𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6540  cz 12554  cuz 12818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-cnex 11162  ax-resscn 11163
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-neg 11443  df-z 12555  df-uz 12819
This theorem is referenced by:  eluzelz2d  44109  uzublem  44126  uzinico  44259  limsupubuzlem  44414  limsupmnfuzlem  44428  limsupre3uzlem  44437  limsupvaluz2  44440  supcnvlimsup  44442  xlimclim2lem  44541  climxlim2  44548  xlimliminflimsup  44564  smflimmpt  45512  smflimsuplem3  45524  smflimsuplem4  45525  smflimsuplem5  45526  smflimsuplem6  45527  smflimsuplem7  45528  smflimsuplem8  45529  smflimsupmpt  45531  smfliminflem  45532  smfliminfmpt  45534
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