Theorem eluzelz2 45968

Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
eluzelz2.1	⊢ 𝑍 = (ℤ≥‘𝑀)

Assertion

Ref	Expression
eluzelz2	⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)

Proof of Theorem eluzelz2

Step	Hyp	Ref	Expression
1		eluzelz2.1	. . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀)
2	1	eleq2i 2855	. . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀))
3	2	biimpi 218	. 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀))
4		eluzelz 12859	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
5	3, 4	syl 17	1 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  ‘cfv 6521  ℤcz 12578  ℤ_≥cuz 12849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-cnex 11140  ax-resscn 11141

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fv 6529  df-ov 7399  df-neg 11428  df-z 12579  df-uz 12850

This theorem is referenced by:  eluzelz2d  45978  uzublem  45995  uzinico  46126  limsupubuzlem  46277  limsupmnfuzlem  46291  limsupre3uzlem  46300  limsupvaluz2  46303  supcnvlimsup  46305  xlimclim2lem  46404  climxlim2  46411  xlimliminflimsup  46427  smflimmpt  47375  smflimsuplem3  47387  smflimsuplem4  47388  smflimsuplem5  47389  smflimsuplem6  47390  smflimsuplem7  47391  smflimsuplem8  47392  smflimsupmpt  47394  smfliminflem  47395  smfliminfmpt  47397