gtnelicc - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > gtnelicc		Structured version Visualization version GIF version

Theorem gtnelicc 44785

Description: A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
gtnelicc.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
gtnelicc.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
gtnelicc.c	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
gtnelicc.bltc	⊢ (𝜑 → 𝐵 < 𝐶)

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

**Proof of Theorem gtnelicc**
Step	Hyp	Ref	Expression
1		gtnelicc.bltc	. . . 4 ⊢ (𝜑 → 𝐵 < 𝐶)
2		gtnelicc.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3	2	rexrd 11268	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
4		gtnelicc.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
5		xrltnle 11285	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
6	3, 4, 5	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝐵 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐵))
7	1, 6	mpbid 231	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝐵)
8	7	intnand 488	. 2 ⊢ (𝜑 → ¬ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵))
9		gtnelicc.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
10		elicc4 13397	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵)))
11	9, 3, 4, 10	syl3anc 1368	. 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 205 ∧ wa 395 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7405 ℝcr 11111 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 [,]cicc 13333

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-xr 11256 df-le 11258 df-icc 13337

This theorem is referenced by: fourierdlem103 45497

W3C validator