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Theorem eluzelz2d 39953
 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 39940 . 2 (𝑁𝑍𝑁 ∈ ℤ)
41, 3syl 17 1 (𝜑𝑁 ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ‘cfv 5926  ℤcz 11415  ℤ≥cuz 11725 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-cnex 10030  ax-resscn 10031 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-neg 10307  df-z 11416  df-uz 11726 This theorem is referenced by:  uzred  39983  limsupequzmpt2  40268  liminfequzmpt2  40341  xlimconst2  40379  smflimsuplem1  41347  smflimsuplem4  41350  smflimsuplem8  41354  smfliminflem  41357
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