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Theorem eluzd 40145
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 11893 . . 3 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
51, 2, 3, 4syl3anbrc 1427 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzd.1 . 2 𝑍 = (ℤ𝑀)
75, 6syl6eleqr 2860 1 (𝜑𝑁𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144   class class class wbr 4784  cfv 6031  cle 10276  cz 11578  cuz 11887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-cnex 10193  ax-resscn 10194
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-neg 10470  df-z 11579  df-uz 11888
This theorem is referenced by:  uzublem  40167  uzinico  40299  uzubioo  40306  limsupubuzlem  40456  limsupequzlem  40466  limsupmnfuzlem  40470  limsupequzmptlem  40472  limsupre3uzlem  40479  supcnvlimsup  40484  limsup10exlem  40516  smflimsuplem4  41543  smfliminflem  41550
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