ltnelicc - Mathbox for Glauco Siliprandi

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Theorem ltnelicc 45886

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 11196	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		xrltnle 11213	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
6	2, 4, 5	syl2anc 585	. . . 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 13343	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	4, 9, 2, 10	syl3anc 1374	. 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 2114 class class class wbr 5100 (class class class)co 7370 ℝcr 11039 ℝ^*cxr 11179 < clt 11180 ≤ cle 11181 [,]cicc 13278

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6458 df-fun 6504 df-fv 6510 df-ov 7373 df-oprab 7374 df-mpo 7375 df-xr 11184 df-le 11186 df-icc 13282

This theorem is referenced by: fourierdlem104 46597

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