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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eluzd | Structured version Visualization version GIF version |
Description: Membership in an upper set of integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
eluzd.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
eluzd.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
eluzd.3 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
eluzd.4 | ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
Ref | Expression |
---|---|
eluzd | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzd.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | eluzd.3 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | eluzd.4 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) | |
4 | eluz2 12882 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) | |
5 | 1, 2, 3, 4 | syl3anbrc 1342 | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
6 | eluzd.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
7 | 5, 6 | eleqtrrdi 2850 | 1 ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 ≤ cle 11294 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: uzublem 45380 uzinico 45513 uzubioo 45520 limsupubuzlem 45668 limsupequzlem 45678 limsupmnfuzlem 45682 limsupequzmptlem 45684 limsupre3uzlem 45691 supcnvlimsup 45696 limsup10exlem 45728 smflimsuplem4 46779 smfliminflem 46786 |
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