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Theorem eluzelz2d 42037
 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 42027 . 2 (𝑁𝑍𝑁 ∈ ℤ)
41, 3syl 17 1 (𝜑𝑁 ∈ ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  ‘cfv 6328  ℤcz 11973  ℤ≥cuz 12235 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-cnex 10586  ax-resscn 10587 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-neg 10866  df-z 11974  df-uz 12236 This theorem is referenced by:  uzred  42067  limsupequzmpt2  42347  liminfequzmpt2  42420  xlimconst2  42464  iundjiunlem  43085  smflimsuplem1  43438  smflimsuplem4  43441  smflimsuplem8  43445  smfliminflem  43448
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