xrgtnelicc - Mathbox for Glauco Siliprandi

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Theorem xrgtnelicc 45805

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 11201	. . . . 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 13331	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	8, 2, 3, 9	syl3anc 1373	. 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 2113 class class class wbr 5098 (class class class)co 7358 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 [,]cicc 13266

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-xr 11172 df-le 11174 df-icc 13270

This theorem is referenced by: iccdificc 45806

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