icogelb - Metamath Proof Explorer

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Theorem icogelb 13338

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 13330	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
2		simp2 1138	. . 3 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → 𝐴 ≤ 𝐶)
3	1, 2	biimtrdi 253	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶))
4	3	3impia 1118	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7358 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 [,)cico 13289

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-iota 6446 df-fun 6492 df-fv 6498 df-ov 7361 df-oprab 7362 df-mpo 7363 df-xr 11172 df-ico 13293

This theorem is referenced by: icogelbd 13339 fprodge0 15947 fprodge1 15949 hgt750lemf 34818 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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