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Theorem eluzd 45425
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 12885 . . 3 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
51, 2, 3, 4syl3anbrc 1343 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzd.1 . 2 𝑍 = (ℤ𝑀)
75, 6eleqtrrdi 2851 1 (𝜑𝑁𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107   class class class wbr 5142  cfv 6560  cle 11297  cz 12615  cuz 12879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-cnex 11212  ax-resscn 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-neg 11496  df-z 12616  df-uz 12880
This theorem is referenced by:  uzublem  45446  uzinico  45578  uzubioo  45585  limsupubuzlem  45732  limsupequzlem  45742  limsupmnfuzlem  45746  limsupequzmptlem  45748  limsupre3uzlem  45755  supcnvlimsup  45760  limsup10exlem  45792  smflimsuplem4  46843  smfliminflem  46850
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