ltnelicc - Mathbox for Glauco Siliprandi

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Theorem ltnelicc 46161

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 11258	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		xrltnle 11275	. . . . 5 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
6	2, 4, 5	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐶 < 𝐴 ↔ ¬ 𝐴 ≤ 𝐶))
7	1, 6	mpbid 235	. . 3 ⊢ (𝜑 → ¬ 𝐴 ≤ 𝐶)
8	7	intnanrd 494	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9		ltnelicc.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
10		elicc4 13439	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	4, 9, 2, 10	syl3anc 1396	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
12	8, 11	mtbird 328	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∈ wcel 2141 class class class wbr 5108 (class class class)co 7410 ℝcr 11098 ℝ^*cxr 11241 < clt 11242 ≤ cle 11243 [,]cicc 13374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5256 ax-pr 5404 ax-un 7732 ax-cnex 11155 ax-resscn 11156

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ral 3078 df-rex 3088 df-rab 3415 df-v 3455 df-sbc 3744 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-iota 6492 df-fun 6538 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-xr 11246 df-le 11248 df-icc 13378

This theorem is referenced by: fourierdlem104 46872

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