ltnelicc - Mathbox for Glauco Siliprandi

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Theorem ltnelicc 39719

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 10089	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		xrltnle 10105	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
6	2, 4, 5	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
7	1, 6	mpbid 222	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶)
8	7	intnanrd 963	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9		ltnelicc.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		elicc4 12240	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	4, 9, 2, 10	syl3anc 1326	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
12	8, 11	mtbird 315	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 [,]cicc 12178

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-xr 10078 df-le 10080 df-icc 12182

This theorem is referenced by: fourierdlem104 40427

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