icogelb - Metamath Proof Explorer

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Theorem icogelb 13340

Description: An element of a left-closed right-open interval is greater than or equal to its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Assertion**
Ref	Expression
icogelb	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)

**Proof of Theorem icogelb**
Step	Hyp	Ref	Expression
1		elico1 13332	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
2		simp2 1143	. . 3 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶)
3	1, 2	biimtrdi 254	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶))
4	3	3impia 1123	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 ∈ wcel 2119 class class class wbr 5072 (class class class)co 7356 ℝ^*cxr 11169 < clt 11170 ≤ cle 11171 [,)cico 13291

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-ico 13295

This theorem is referenced by: icogelbd 13341 fprodge0 15949 fprodge1 15951 hgt750lemf 34837 xralrple2 45799 icoopn 45970 fsumge0cl 46018 limcresioolb 46086 fourierdlem41 46591 fourierdlem43 46593 fourierdlem46 46595 fourierdlem48 46597 fouriersw 46674 sge0isum 46870 sge0ad2en 46874 sge0uzfsumgt 46887 sge0seq 46889 sge0reuz 46890 hoidmv1lelem2 47035 hoidmvlelem1 47038 hoidmvlelem2 47039 ovnhoilem1 47044 hspdifhsp 47059 hspmbllem2 47070 iinhoiicclem 47116

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