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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 12774 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
7 | 4, 5, 6 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
8 | 1, 2, 3, 7 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 (,]cioc 12733 |
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-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-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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-xr 10673 df-ioc 12737 |
This theorem is referenced by: iocopn 41789 eliccelioc 41790 iccdificc 41808 ressiocsup 41823 iooiinioc 41825 preimaiocmnf 41830 xlimpnfvlem2 42111 ioccncflimc 42161 fourierdlem41 42427 fourierdlem46 42431 fourierdlem48 42433 fourierdlem49 42434 fourierdlem51 42436 fourierswlem 42509 smfsuplem1 43079 |
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