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Theorem infxrss 13260

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 768	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ℝ^)
2		simpl 482	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) → 𝐴 ⊆ 𝐵)
3	2	sselda 3937	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^*) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
4		infxrlb 13255	. . . 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 3121	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → ∀𝑥 ∈ 𝐴 inf(𝐵, ℝ^, < ) ≤ 𝑥)
7		sstr 3946	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
8		infxrcl 13254	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → inf(𝐵, ℝ^, < ) ∈ ℝ^)
9	8	adantl 481	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → inf(𝐵, ℝ^, < ) ∈ ℝ^*)
10		infxrgelb 13256	. . 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 257	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ^) → inf(𝐵, ℝ^, < ) ≤ inf(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3905 class class class wbr 5095 infcinf 9350 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169

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 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-po 5531 df-so 5532 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368

This theorem is referenced by: infxrpnf 45426 ioossioobi 45499 liminflelimsuplem 45757 ovnsslelem 46542 ovolval5lem3 46636

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