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Theorem eluzd 39948
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 11731 . . 3 (𝑁 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀𝑁))
51, 2, 3, 4syl3anbrc 1265 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
6 eluzd.1 . 2 𝑍 = (ℤ𝑀)
75, 6syl6eleqr 2741 1 (𝜑𝑁𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  cle 10113  cz 11415  cuz 11725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-neg 10307  df-z 11416  df-uz 11726
This theorem is referenced by:  uzublem  39970  uzinico  40105  uzubioo  40112  limsupubuzlem  40262  limsupequzlem  40272  limsupmnfuzlem  40276  limsupequzmptlem  40278  limsupre3uzlem  40285  supcnvlimsup  40290  limsup10exlem  40322  smflimsuplem4  41350  smfliminflem  41357
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