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Theorem eluzd 41558
Description: Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
eluzd.1 𝑍 = (ℤ𝑀)
eluzd.2 (𝜑𝑀 ∈ ℤ)
eluzd.3 (𝜑𝑁 ∈ ℤ)
eluzd.4 (𝜑𝑀𝑁)
Assertion
Ref Expression
eluzd (𝜑𝑁𝑍)

Proof of Theorem eluzd
StepHypRef Expression
1 eluzd.2 . . 3 (𝜑𝑀 ∈ ℤ)
2 eluzd.3 . . 3 (𝜑𝑁 ∈ ℤ)
3 eluzd.4 . . 3 (𝜑𝑀𝑁)
4 eluz2 12237 . . 3 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
51, 2, 3, 4syl3anbrc 1335 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzd.1 . 2 𝑍 = (ℤ𝑀)
75, 6eleqtrrdi 2921 1 (𝜑𝑁𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  cle 10664  cz 11969  cuz 12231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-neg 10861  df-z 11970  df-uz 12232
This theorem is referenced by:  uzublem  41580  uzinico  41712  uzubioo  41719  limsupubuzlem  41869  limsupequzlem  41879  limsupmnfuzlem  41883  limsupequzmptlem  41885  limsupre3uzlem  41892  supcnvlimsup  41897  limsup10exlem  41929  smflimsuplem4  42974  smfliminflem  42981
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