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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 11245 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
2 | 0nonelaleb.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 0nonelalab.5 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
4 | elioore 13384 | . . . 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 11292 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
8 | 0nonelaleb.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | 8 | rexrd 11292 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
10 | elioo2 13395 | . . . . . 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 11403 | . 2 ⊢ (𝜑 → 0 < 𝐶) |
15 | 1, 14 | ltned 11378 | 1 ⊢ (𝜑 → 0 ≠ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2930 class class class wbr 5143 (class class class)co 7415 ℝcr 11135 0cc0 11136 ℝ*cxr 11275 < clt 11276 ≤ cle 11277 (,)cioo 13354 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-addrcl 11197 ax-rnegex 11207 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7989 df-2nd 7990 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-ioo 13358 |
This theorem is referenced by: dvrelogpow2b 41594 |
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