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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lenelioc | Structured version Visualization version GIF version |
Description: A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
lenelioc.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
lenelioc.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
lenelioc.3 | ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
lenelioc.4 | ⊢ (𝜑 → 𝐶 ≤ 𝐴) |
Ref | Expression |
---|---|
lenelioc | ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lenelioc.4 | . . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴) | |
2 | lenelioc.3 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | |
3 | lenelioc.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
4 | 2, 3 | xrlenltd 11320 | . . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶)) |
5 | 1, 4 | mpbid 231 | . . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶) |
6 | 5 | intn3an2d 1476 | . 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) |
7 | lenelioc.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) | |
8 | elioc1 13408 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) | |
9 | 3, 7, 8 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))) |
10 | 6, 9 | mtbird 324 | 1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 ℝ*cxr 11287 < clt 11288 ≤ cle 11289 (,]cioc 13367 |
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-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-xr 11292 df-le 11294 df-ioc 13371 |
This theorem is referenced by: (None) |
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