xrgtnelicc - Mathbox for Glauco Siliprandi

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Theorem xrgtnelicc 43305

Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

**Hypotheses**
Ref	Expression
xrgtnelicc.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xrgtnelicc.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
xrgtnelicc.3	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
xrgtnelicc.4	⊢ (𝜑 → 𝐵 < 𝐶)

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

**Proof of Theorem xrgtnelicc**
Step	Hyp	Ref	Expression
1		xrgtnelicc.4	. . . 4 ⊢ (𝜑 → 𝐵 < 𝐶)
2		xrgtnelicc.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		xrgtnelicc.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
4		xrltnle 11092	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
5	2, 3, 4	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
6	1, 5	mpbid 231	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵)
7	6	intnand 490	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
8		xrgtnelicc.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
9		elicc4 13196	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	8, 2, 3, 9	syl3anc 1371	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	7, 10	mtbird 325	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2104 class class class wbr 5081 (class class class)co 7307 ℝ^*cxr 11058 < clt 11059 ≤ cle 11060 [,]cicc 13132

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-un 7620 ax-cnex 10977 ax-resscn 10978

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3306 df-v 3439 df-sbc 3722 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-iota 6410 df-fun 6460 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-xr 11063 df-le 11065 df-icc 13136

This theorem is referenced by: iccdificc 43306

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