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Theorem infxrgelb 13346

Description: The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)

**Assertion**
Ref	Expression
infxrgelb	⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem infxrgelb Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrltso 13152	. . . . . 6 ⊢ < Or ℝ^*
2	1	a1i 11	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
3		xrinfmss 13321	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → ∃𝑧 ∈ ℝ^* (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑧 ∧ ∀𝑦 ∈ ℝ^* (𝑧 < 𝑦 → ∃𝑥 ∈ 𝐴 𝑥 < 𝑦)))
4		id 22	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → 𝐴 ⊆ ℝ^*)
5	2, 3, 4	infglbb 9514	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (inf(𝐴, ℝ^, < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵))
6	5	notbid 317	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ inf(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵))
7		ralnex 3062	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)
8	6, 7	bitr4di 288	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ inf(𝐴, ℝ^, < ) < 𝐵 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵))
9		id 22	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ∈ ℝ^*)
10		infxrcl 13344	. . 3 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
11		xrlenlt 11309	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ inf(𝐴, ℝ^, < ) ∈ ℝ^) → (𝐵 ≤ inf(𝐴, ℝ^, < ) ↔ ¬ inf(𝐴, ℝ^, < ) < 𝐵))
12	9, 10, 11	syl2anr 595	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ≤ inf(𝐴, ℝ^, < ) ↔ ¬ inf(𝐴, ℝ^*, < ) < 𝐵))
13		simplr 767	. . . 4 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^)
14		simpl 481	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ⊆ ℝ^)
15	14	sselda 3972	. . . 4 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
16	13, 15	xrlenltd 11310	. . 3 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵))
17	16	ralbidva 3166	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵))
18	8, 12, 17	3bitr4d 310	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 ⊆ wss 3939 class class class wbr 5143 Or wor 5583 infcinf 9464 ℝ^*cxr 11277 < clt 11278 ≤ cle 11279

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477

This theorem is referenced by: infxrre 13347 infxrss 13350 ixxlb 13378 limsuple 15454 limsupval2 15456 imasdsf1olem 24297 nmogelb 24651 metdsf 24782 metdsge 24783 ovolgelb 25427 ovolge0 25428 ovolsslem 25431 ovolicc2 25469 ismblfin 37191 infrpge 44796 infleinf2 44859 infxrgelbrnmpt 44899 inficc 44982 liminfgord 45205 liminflelimsuplem 45226 ovnhoilem2 46053

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