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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliocd | Structured version Visualization version GIF version |
Description: Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
eliocd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
eliocd.b | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
eliocd.c | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
eliocd.altc | ⊢ (𝜑 → 𝐴 < 𝐶) |
eliocd.cleb | ⊢ (𝜑 → 𝐶 ≤ 𝐵) |
Ref | Expression |
---|---|
eliocd | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliocd.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
2 | eliocd.altc | . 2 ⊢ (𝜑 → 𝐴 < 𝐶) | |
3 | eliocd.cleb | . 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵) | |
4 | eliocd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
5 | eliocd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
6 | elioc1 13426 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
7 | 4, 5, 6 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1341 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 (,]cioc 13385 |
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-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-sbc 3792 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-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11297 df-ioc 13389 |
This theorem is referenced by: iocopn 45473 eliccelioc 45474 iccdificc 45492 ressiocsup 45507 iooiinioc 45509 preimaiocmnf 45514 xlimpnfvlem2 45793 ioccncflimc 45841 fourierdlem41 46104 fourierdlem46 46108 fourierdlem48 46110 fourierdlem49 46111 fourierdlem51 46113 fourierswlem 46186 smfsuplem1 46767 |
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