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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 12768 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
7 | 4, 5, 6 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1339 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 (,]cioc 12727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-xr 10668 df-ioc 12731 |
This theorem is referenced by: iocopn 42157 eliccelioc 42158 iccdificc 42176 ressiocsup 42191 iooiinioc 42193 preimaiocmnf 42198 xlimpnfvlem2 42479 ioccncflimc 42527 fourierdlem41 42790 fourierdlem46 42794 fourierdlem48 42796 fourierdlem49 42797 fourierdlem51 42799 fourierswlem 42872 smfsuplem1 43442 |
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