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Theorem infxrgelb 13303

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 13108	. . . . . 6 ⊢ < Or ℝ^*
2	1	a1i 11	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → < Or ℝ^*)
3		xrinfmss 13277	. . . . 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 9450	. . . 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 3056	. . 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 13301	. . 3 ⊢ (𝐴 ⊆ ℝ^* → inf(𝐴, ℝ^, < ) ∈ ℝ^)
11		xrlenlt 11246	. . 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 3949	. . . 4 ⊢ (((𝐴 ⊆ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℝ^)
16	13, 15	xrlenltd 11247	. . 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 2109 ∀wral 3045 ∃wrex 3054 ⊆ wss 3917 class class class wbr 5110 Or wor 5548 infcinf 9399 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415

This theorem is referenced by: infxrre 13304 infxrss 13307 ixxlb 13335 limsuple 15451 limsupval2 15453 imasdsf1olem 24268 nmogelb 24611 metdsf 24744 metdsge 24745 ovolgelb 25388 ovolge0 25389 ovolsslem 25392 ovolicc2 25430 ismblfin 37662 infrpge 45354 infleinf2 45417 infxrgelbrnmpt 45457 inficc 45539 liminfgord 45759 liminflelimsuplem 45780 ovnhoilem2 46607

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