lenelioc - Mathbox for Glauco Siliprandi

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Theorem lenelioc 46073

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 11242	. . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝐴 ↔ ¬ 𝐴 < 𝐶))
5	1, 4	mpbid 234	. . 3 ⊢ (𝜑 → ¬ 𝐴 < 𝐶)
6	5	intn3an2d 1500	. 2 ⊢ (𝜑 → ¬ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
7		lenelioc.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
8		elioc1 13385	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
9	3, 7, 8	syl2anc 593	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
10	6, 9	mtbird 327	1 ⊢ (𝜑 → ¬ 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ w3a 1097 ∈ wcel 2141 class class class wbr 5097 (class class class)co 7391 ℝ^*cxr 11209 < clt 11210 ≤ cle 11211 (,]cioc 13344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-br 5098 df-opab 5160 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-iota 6472 df-fun 6518 df-fv 6524 df-ov 7394 df-oprab 7395 df-mpo 7396 df-xr 11214 df-le 11216 df-ioc 13348

This theorem is referenced by: (None)

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