xrgtnelicc - Mathbox for Glauco Siliprandi

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Theorem xrgtnelicc 45556

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 11329	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
6	1, 5	mpbid 232	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵)
7	6	intnand 488	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
8		xrgtnelicc.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
9		elicc4 13455	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	8, 2, 3, 9	syl3anc 1372	. 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 206 ∧ wa 395 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 ℝ^*cxr 11295 < clt 11296 ≤ cle 11297 [,]cicc 13391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-xr 11300 df-le 11302 df-icc 13395

This theorem is referenced by: iccdificc 45557

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