icogelb - Metamath Proof Explorer

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Theorem icogelb 13435

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 13427	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
2		simp2 1155	. . 3 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶)
3	1, 2	biimtrdi 256	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶))
4	3	3impia 1135	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2146 class class class wbr 5111 (class class class)co 7416 ℝ^*cxr 11253 < clt 11254 ≤ cle 11255 [,)cico 13386

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-iota 6496 df-fun 6542 df-fv 6548 df-ov 7419 df-oprab 7420 df-mpo 7421 df-xr 11258 df-ico 13390

This theorem is used by: icogelbd 13436 fprodge0 16066 fprodge1 16068 hgt750lemf 35081 xralrple2 46103 icoopn 46274 fsumge0cl 46322 limcresioolb 46390 fourierdlem41 46895 fourierdlem43 46897 fourierdlem46 46899 fouriersw 46978 sge0isum 47174 sge0ad2en 47178 sge0uzfsumgt 47191 sge0seq 47193 sge0reuz 47194 hoidmv1lelem2 47339 hoidmvlelem1 47342 hoidmvlelem2 47343 ovnhoilem1 47348 hspdifhsp 47363 hspmbllem2 47374 iinhoiicclem 47420

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