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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 0nonelalab | Structured version Visualization version GIF version | ||
| Description: Technical lemma for open interval. (Contributed by metakunt, 12-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| 0nonelaleb.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| 0nonelaleb.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) | 
| 0nonelaleb.3 | ⊢ (𝜑 → 0 < 𝐴) | 
| 0nonelaleb.4 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) | 
| 0nonelalab.5 | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | 
| Ref | Expression | 
|---|---|
| 0nonelalab | ⊢ (𝜑 → 0 ≠ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0red 11264 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
| 2 | 0nonelaleb.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 3 | 0nonelalab.5 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
| 4 | elioore 13417 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 ∈ ℝ) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| 6 | 0nonelaleb.3 | . . 3 ⊢ (𝜑 → 0 < 𝐴) | |
| 7 | 2 | rexrd 11311 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | 
| 8 | 0nonelaleb.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
| 9 | 8 | rexrd 11311 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | 
| 10 | elioo2 13428 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
| 11 | 7, 9, 10 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | 
| 12 | 3, 11 | mpbid 232 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) | 
| 13 | 12 | simp2d 1144 | . . 3 ⊢ (𝜑 → 𝐴 < 𝐶) | 
| 14 | 1, 2, 5, 6, 13 | lttrd 11422 | . 2 ⊢ (𝜑 → 0 < 𝐶) | 
| 15 | 1, 14 | ltned 11397 | 1 ⊢ (𝜑 → 0 ≠ 𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-addrcl 11216 ax-rnegex 11226 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 | 
| This theorem is referenced by: dvrelogpow2b 42069 | 
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