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Theorem infxrss 13314

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 767	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ℝ^)
2		simpl 483	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) → 𝐴 ⊆ 𝐵)
3	2	sselda 3981	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
4		infxrlb 13309	. . . 4 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝑥 ∈ 𝐵) → inf(𝐵, ℝ^*, < ) ≤ 𝑥)
5	1, 3, 4	syl2anc 584	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → inf(𝐵, ℝ^, < ) ≤ 𝑥)
6	5	ralrimiva 3146	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → ∀𝑥 ∈ 𝐴 inf(𝐵, ℝ^, < ) ≤ 𝑥)
7		sstr 3989	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
8		infxrcl 13308	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → inf(𝐵, ℝ^, < ) ∈ ℝ^)
9	8	adantl 482	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
10		infxrgelb 13310	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ inf(𝐵, ℝ^, < ) ∈ ℝ^) → (inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 inf(𝐵, ℝ^*, < ) ≤ 𝑥))
11	7, 9, 10	syl2anc 584	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → (inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^, < ) ↔ ∀𝑥 ∈ 𝐴 inf(𝐵, ℝ^, < ) ≤ 𝑥))
12	6, 11	mpbird 256	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3061 ⊆ wss 3947 class class class wbr 5147 infcinf 9432 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443

This theorem is referenced by: infxrpnf 44142 ioossioobi 44216 liminflelimsuplem 44477 ovnsslelem 45262 ovolval5lem3 45356

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