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Theorem eluzelz2 39940
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 2722 . . 3 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
32biimpi 206 . 2 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
4 eluzelz 11735 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
53, 4syl 17 1 (𝑁𝑍𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cfv 5926  cz 11415  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:  eluzelz2d  39953  uzublem  39970  uzinico  40105  limsupubuzlem  40262  limsupmnfuzlem  40276  limsupre3uzlem  40285  limsupvaluz2  40288  supcnvlimsup  40290  xlimclim2lem  40383  climxlim2  40390  smflimmpt  41337  smflimsuplem3  41349  smflimsuplem4  41350  smflimsuplem5  41351  smflimsuplem6  41352  smflimsuplem7  41353  smflimsuplem8  41354  smflimsupmpt  41356  smfliminflem  41357  smfliminfmpt  41359
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