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Theorem eluzelz2 40125
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 206 . 2 (𝑁𝑍𝑁 ∈ (ℤ𝑀))
4 eluzelz 11889 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
53, 4syl 17 1 (𝑁𝑍𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  cfv 6049  cz 11569  cuz 11879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-neg 10461  df-z 11570  df-uz 11880
This theorem is referenced by:  eluzelz2d  40138  uzublem  40155  uzinico  40290  limsupubuzlem  40447  limsupmnfuzlem  40461  limsupre3uzlem  40470  limsupvaluz2  40473  supcnvlimsup  40475  xlimclim2lem  40568  climxlim2  40575  smflimmpt  41522  smflimsuplem3  41534  smflimsuplem4  41535  smflimsuplem5  41536  smflimsuplem6  41537  smflimsuplem7  41538  smflimsuplem8  41539  smflimsupmpt  41541  smfliminflem  41542  smfliminfmpt  41544
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