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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 10715 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | 0nonelaleb.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 0nonelalab.5 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
4 | elioore 12844 | . . . 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 10762 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | 0nonelaleb.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | rexrd 10762 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
10 | elioo2 12855 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
11 | 7, 9, 10 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
12 | 3, 11 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) |
13 | 12 | simp2d 1144 | . . 3 ⊢ (𝜑 → 𝐴 < 𝐶) |
14 | 1, 2, 5, 6, 13 | lttrd 10872 | . 2 ⊢ (𝜑 → 0 < 𝐶) |
15 | 1, 14 | ltned 10847 | 1 ⊢ (𝜑 → 0 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1088 ∈ wcel 2113 ≠ wne 2934 class class class wbr 5027 (class class class)co 7164 ℝcr 10607 0cc0 10608 ℝ*cxr 10745 < clt 10746 ≤ cle 10747 (,)cioo 12814 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-addrcl 10669 ax-rnegex 10679 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-ioo 12818 |
This theorem is referenced by: dvrelogpow2b 39684 |
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