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Theorem eluzelz2d 44109
Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
eluzelz2d.1 𝑍 = (ℤ𝑀)
eluzelz2d.2 (𝜑𝑁𝑍)
Assertion
Ref Expression
eluzelz2d (𝜑𝑁 ∈ ℤ)

Proof of Theorem eluzelz2d
StepHypRef Expression
1 eluzelz2d.2 . 2 (𝜑𝑁𝑍)
2 eluzelz2d.1 . . 3 𝑍 = (ℤ𝑀)
32eluzelz2 44099 . 2 (𝑁𝑍𝑁 ∈ ℤ)
41, 3syl 17 1 (𝜑𝑁 ∈ 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:  uzred  44139  cvgcau  44187  limsupequzmpt2  44420  liminfequzmpt2  44493  xlimconst2  44537  iundjiunlem  45161  smflimsuplem1  45522  smflimsuplem4  45525  smflimsuplem8  45529  smfliminflem  45532
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