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Theorem eluzelzd 44792
Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
eluzelzd.1 (𝜑𝑁 ∈ (ℤ𝑀))
Assertion
Ref Expression
eluzelzd (𝜑𝑁 ∈ ℤ)

Proof of Theorem eluzelzd
StepHypRef Expression
1 eluzelzd.1 . 2 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzelz 12860 . 2 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
31, 2syl 17 1 (𝜑𝑁 ∈ ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  cfv 6541  cz 12586  cuz 12850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pr 5421  ax-cnex 11192  ax-resscn 11193
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7417  df-neg 11475  df-z 12587  df-uz 12851
This theorem is referenced by:  meaiininclem  45909
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