lenelioc - Mathbox for Glauco Siliprandi

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Theorem lenelioc 45981

Description: A real number smaller than or equal to the lower bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

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

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

**Proof of Theorem lenelioc**
Step	Hyp	Ref	Expression
1		lenelioc.4	. . . 4 ⊢ (𝜑 → 𝐶 ≤ 𝐴)
2		lenelioc.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
3		lenelioc.1	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
4	2, 3	xrlenltd 11202	. . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶))
5	1, 4	mpbid 233	. . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶)
6	5	intn3an2d 1488	. 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
7		lenelioc.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
8		elioc1 13331	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
9	3, 7, 8	syl2anc 590	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	6, 9	mtbird 326	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ w3a 1092 ∈ wcel 2119 class class class wbr 5072 (class class class)co 7356 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 (,]cioc 13290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-iota 6441 df-fun 6487 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-xr 11174 df-le 11176 df-ioc 13294

This theorem is referenced by: (None)

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