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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 11255 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | 0nonelaleb.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 0nonelalab.5 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
4 | elioore 13394 | . . . 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 11302 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | 0nonelaleb.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | rexrd 11302 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
10 | elioo2 13405 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
11 | 7, 9, 10 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
12 | 3, 11 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) |
13 | 12 | simp2d 1140 | . . 3 ⊢ (𝜑 → 𝐴 < 𝐶) |
14 | 1, 2, 5, 6, 13 | lttrd 11413 | . 2 ⊢ (𝜑 → 0 < 𝐶) |
15 | 1, 14 | ltned 11388 | 1 ⊢ (𝜑 → 0 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 (class class class)co 7426 ℝcr 11145 0cc0 11146 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 (,)cioo 13364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-addrcl 11207 ax-rnegex 11217 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-ioo 13368 |
This theorem is referenced by: dvrelogpow2b 41571 |
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