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Theorem eluzd 43730
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 12774 . . 3 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
51, 2, 3, 4syl3anbrc 1344 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzd.1 . 2 𝑍 = (ℤ𝑀)
75, 6eleqtrrdi 2845 1 (𝜑𝑁𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107   class class class wbr 5106  cfv 6497  cle 11195  cz 12504  cuz 12768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-cnex 11112  ax-resscn 11113
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-neg 11393  df-z 12505  df-uz 12769
This theorem is referenced by:  uzublem  43751  uzinico  43884  uzubioo  43891  limsupubuzlem  44039  limsupequzlem  44049  limsupmnfuzlem  44053  limsupequzmptlem  44055  limsupre3uzlem  44062  supcnvlimsup  44067  limsup10exlem  44099  smflimsuplem4  45150  smfliminflem  45157
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