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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliccxrd | Structured version Visualization version GIF version |
Description: Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
eliccxrd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
eliccxrd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
eliccxrd.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
eliccxrd.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐶) |
eliccxrd.5 | ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Ref | Expression |
---|---|
eliccxrd | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccxrd.4 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐶) | |
2 | eliccxrd.5 | . . 3 ⊢ (𝜑 → 𝐶 ≤ 𝐵) | |
3 | 1, 2 | jca 512 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)) |
4 | eliccxrd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | eliccxrd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
6 | eliccxrd.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
7 | elicc4 12791 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
8 | 4, 5, 6, 7 | syl3anc 1363 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))) |
9 | 3, 8 | mpbird 258 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℝ*cxr 10662 ≤ cle 10664 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-xr 10667 df-icc 12733 |
This theorem is referenced by: inficc 41686 iccdificc 41691 sge0cl 42540 sge0p1 42573 sge0rpcpnf 42580 ovnsubaddlem1 42729 ovolval5lem1 42811 |
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