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Theorem eluzelz2 45353
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 2831 . . 3 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
32biimpi 216 . 2 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
4 eluzelz 12886 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
53, 4syl 17 1 (𝑁𝑍𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  cz 12611  cuz 12876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-neg 11493  df-z 12612  df-uz 12877
This theorem is referenced by:  eluzelz2d  45363  uzublem  45380  uzinico  45513  limsupubuzlem  45668  limsupmnfuzlem  45682  limsupre3uzlem  45691  limsupvaluz2  45694  supcnvlimsup  45696  xlimclim2lem  45795  climxlim2  45802  xlimliminflimsup  45818  smflimmpt  46766  smflimsuplem3  46778  smflimsuplem4  46779  smflimsuplem5  46780  smflimsuplem6  46781  smflimsuplem7  46782  smflimsuplem8  46783  smflimsupmpt  46785  smfliminflem  46786  smfliminfmpt  46788
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