icogelb - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > icogelb		Structured version Visualization version GIF version

Theorem icogelb 13567

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 13559	. . 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 2147 class class class wbr 5108 (class class class)co 7451 ℝ^*cxr 11381 < clt 11382 ≤ cle 11383 [,)cico 13518

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-iota 6520 df-fun 6568 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-xr 11386 df-ico 13522

This theorem is used by: icogelbd 13568 fprodge0 16204 fprodge1 16206 hgt750lemf 35393 xralrple2 46639 icoopn 46810 fsumge0cl 46858 limcresioolb 46926 fourierdlem41 47431 fourierdlem43 47433 fourierdlem46 47435 fouriersw 47514 sge0isum 47710 sge0ad2en 47714 sge0uzfsumgt 47727 sge0seq 47729 sge0reuz 47730 hoidmv1lelem2 47875 hoidmvlelem1 47878 hoidmvlelem2 47879 ovnhoilem1 47884 hspdifhsp 47899 hspmbllem2 47910 iinhoiicclem 47956

W3C validator