ltnelicc - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ltnelicc		Structured version Visualization version GIF version

Theorem ltnelicc 45467

Description: A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
ltnelicc.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ltnelicc.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
ltnelicc.c	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
ltnelicc.clta	⊢ (𝜑 → 𝐶 < 𝐴)

**Assertion**
Ref	Expression
ltnelicc	⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

**Proof of Theorem ltnelicc**
Step	Hyp	Ref	Expression
1		ltnelicc.clta	. . . 4 ⊢ (𝜑 → 𝐶 < 𝐴)
2		ltnelicc.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
3		ltnelicc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4	3	rexrd 11293	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		xrltnle 11310	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
6	2, 4, 5	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
7	1, 6	mpbid 232	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶)
8	7	intnanrd 489	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9		ltnelicc.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		elicc4 13436	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	4, 9, 2, 10	syl3anc 1372	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
12	8, 11	mtbird 325	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 class class class wbr 5123 (class class class)co 7413 ℝcr 11136 ℝ^*cxr 11276 < clt 11277 ≤ cle 11278 [,]cicc 13372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ral 3051 df-rex 3060 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4888 df-br 5124 df-opab 5186 df-id 5558 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-iota 6494 df-fun 6543 df-fv 6549 df-ov 7416 df-oprab 7417 df-mpo 7418 df-xr 11281 df-le 11283 df-icc 13376

This theorem is referenced by: fourierdlem104 46182

W3C validator