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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eluzelz2 | Structured version Visualization version GIF version |
Description: A member of an upper set of integers is an integer. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
eluzelz2.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
eluzelz2 | ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz2.1 | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | 1 | eleq2i 2831 | . . 3 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ≥‘𝑀)) |
3 | 2 | biimpi 216 | . 2 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | eluzelz 12886 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 ℤcz 12611 ℤ≥cuz 12876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-neg 11493 df-z 12612 df-uz 12877 |
This theorem is referenced by: eluzelz2d 45363 uzublem 45380 uzinico 45513 limsupubuzlem 45668 limsupmnfuzlem 45682 limsupre3uzlem 45691 limsupvaluz2 45694 supcnvlimsup 45696 xlimclim2lem 45795 climxlim2 45802 xlimliminflimsup 45818 smflimmpt 46766 smflimsuplem3 46778 smflimsuplem4 46779 smflimsuplem5 46780 smflimsuplem6 46781 smflimsuplem7 46782 smflimsuplem8 46783 smflimsupmpt 46785 smfliminflem 46786 smfliminfmpt 46788 |
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