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Theorem infxrss 12362

Description: Larger sets of extended reals have smaller infima. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)

**Assertion**
Ref	Expression
infxrss	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^*, < ))

Proof of Theorem infxrss Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 809	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ℝ^)
2		simpl 474	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) → 𝐴 ⊆ 𝐵)
3	2	sselda 3744	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
4		infxrlb 12357	. . . 4 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐵) → inf(𝐵, ℝ^*, < ) ≤ 𝑥)
5	1, 3, 4	syl2anc 696	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → inf(𝐵, ℝ^, < ) ≤ 𝑥)
6	5	ralrimiva 3104	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → ∀𝑥 ∈ 𝐴 inf(𝐵, ℝ^, < ) ≤ 𝑥)
7		sstr 3752	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
8		infxrcl 12356	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → inf(𝐵, ℝ^, < ) ∈ ℝ^)
9	8	adantl 473	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
10		infxrgelb 12358	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ inf(𝐵, ℝ^, < ) ∈ ℝ^) → (inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 inf(𝐵, ℝ^*, < ) ≤ 𝑥))
11	7, 9, 10	syl2anc 696	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → (inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 inf(𝐵, ℝ^, < ) ≤ 𝑥))
12	6, 11	mpbird 247	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2139 ∀wral 3050 ⊆ wss 3715 class class class wbr 4804 infcinf 8512 ℝ^*cxr 10265 < clt 10266 ≤ cle 10267

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-po 5187 df-so 5188 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461

This theorem is referenced by: infxrpnf 40172 ioossioobi 40246 liminflelimsuplem 40510 ovnsslelem 41280 ovolval5lem3 41374

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