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Theorem infxrgelb 13338

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 13144	. . . . . 6 ⊢ < Or ℝ^*
2	1	a1i 11	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
3		xrinfmss 13313	. . . . 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 9506	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (inf(𝐴, ℝ^, < ) < 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵))
6	5	notbid 318	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ inf(𝐴, ℝ^, < ) < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵))
7		ralnex 3067	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ↔ ¬ ∃𝑥 ∈ 𝐴 𝑥 < 𝐵)
8	6, 7	bitr4di 289	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (¬ inf(𝐴, ℝ^, < ) < 𝐵 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵))
9		id 22	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ∈ ℝ^*)
10		infxrcl 13336	. . 3 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
11		xrlenlt 11301	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ inf(𝐴, ℝ^, < ) ∈ ℝ^) → (𝐵 ≤ inf(𝐴, ℝ^, < ) ↔ ¬ inf(𝐴, ℝ^, < ) < 𝐵))
12	9, 10, 11	syl2anr 596	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ≤ inf(𝐴, ℝ^, < ) ↔ ¬ inf(𝐴, ℝ^*, < ) < 𝐵))
13		simplr 768	. . . 4 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ^)
14		simpl 482	. . . . 5 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ⊆ ℝ^)
15	14	sselda 3978	. . . 4 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
16	13, 15	xrlenltd 11302	. . 3 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵))
17	16	ralbidva 3170	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵))
18	8, 12, 17	3bitr4d 311	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 395 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 ⊆ wss 3944 class class class wbr 5142 Or wor 5583 infcinf 9456 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 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 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469

This theorem is referenced by: infxrre 13339 infxrss 13342 ixxlb 13370 limsuple 15446 limsupval2 15448 imasdsf1olem 24266 nmogelb 24620 metdsf 24751 metdsge 24752 ovolgelb 25396 ovolge0 25397 ovolsslem 25400 ovolicc2 25438 ismblfin 37069 infrpge 44656 infleinf2 44719 infxrgelbrnmpt 44759 inficc 44842 liminfgord 45065 liminflelimsuplem 45086 ovnhoilem2 45913

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