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Theorem infxrgelb 13249

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 13053	. . . . . 6 ⊢ < Or ℝ^*
2	1	a1i 11	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
3		xrinfmss 13223	. . . . 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 9393	. . . 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 3060	. . 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 13247	. . 3 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
11		xrlenlt 11195	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ inf(𝐴, ℝ^, < ) ∈ ℝ^) → (𝐵 ≤ inf(𝐴, ℝ^, < ) ↔ ¬ inf(𝐴, ℝ^, < ) < 𝐵))
12	9, 10, 11	syl2anr 597	. 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 3931	. . . 4 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
16	13, 15	xrlenltd 11196	. . 3 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝑥 ∈ 𝐴) → (𝐵 ≤ 𝑥 ↔ ¬ 𝑥 < 𝐵))
17	16	ralbidva 3155	. 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 206 ∧ wa 395 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 ⊆ wss 3899 class class class wbr 5096 Or wor 5529 infcinf 9342 ℝ^*cxr 11163 < clt 11164 ≤ cle 11165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-po 5530 df-so 5531 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8633 df-en 8882 df-dom 8883 df-sdom 8884 df-sup 9343 df-inf 9344 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365

This theorem is referenced by: infxrre 13250 infxrss 13253 ixxlb 13281 limsuple 15399 limsupval2 15401 imasdsf1olem 24315 nmogelb 24658 metdsf 24791 metdsge 24792 ovolgelb 25435 ovolge0 25436 ovolsslem 25439 ovolicc2 25477 ismblfin 37801 infrpge 45538 infleinf2 45600 infxrgelbrnmpt 45640 inficc 45722 liminfgord 45940 liminflelimsuplem 45961 ovnhoilem2 46788

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