ltnelicc - Mathbox for Glauco Siliprandi

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Theorem ltnelicc 45954

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 11191	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		xrltnle 11208	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
6	2, 4, 5	syl2anc 591	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
7	1, 6	mpbid 234	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶)
8	7	intnanrd 491	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9		ltnelicc.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		elicc4 13361	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	4, 9, 2, 10	syl3anc 1380	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
12	8, 11	mtbird 327	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∈ wcel 2121 class class class wbr 5074 (class class class)co 7359 ℝcr 11033 ℝ^*cxr 11174 < clt 11175 ≤ cle 11176 [,]cicc 13296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5220 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3725 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-br 5075 df-opab 5137 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6444 df-fun 6490 df-fv 6496 df-ov 7362 df-oprab 7363 df-mpo 7364 df-xr 11179 df-le 11181 df-icc 13300

This theorem is referenced by: fourierdlem104 46665

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