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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfzd | Structured version Visualization version GIF version |
Description: Membership in a finite set of sequential integers. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
elfzd.1 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
elfzd.2 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
elfzd.3 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
elfzd.4 | ⊢ (𝜑 → 𝑀 ≤ 𝐾) |
elfzd.5 | ⊢ (𝜑 → 𝐾 ≤ 𝑁) |
Ref | Expression |
---|---|
elfzd | ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzd.1 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | elfzd.2 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | elfzd.3 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
4 | 1, 2, 3 | 3jca 1124 | . . 3 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ)) |
5 | elfzd.4 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝐾) | |
6 | elfzd.5 | . . 3 ⊢ (𝜑 → 𝐾 ≤ 𝑁) | |
7 | 4, 5, 6 | jca32 518 | . 2 ⊢ (𝜑 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
8 | elfz2 12893 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
9 | 7, 8 | sylibr 236 | 1 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ≤ cle 10670 ℤcz 11975 ...cfz 12886 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-neg 10867 df-z 11976 df-fz 12887 |
This theorem is referenced by: uzublem 41697 limsupubuzlem 41986 |
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