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Mirrors > Home > MPE Home > Th. List > supxrss | Structured version Visualization version GIF version |
Description: Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.) |
Ref | Expression |
---|---|
supxrss | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 766 | . . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ ℝ*) | |
2 | simpl 482 | . . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → 𝐴 ⊆ 𝐵) | |
3 | 2 | sselda 3975 | . . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) |
4 | supxrub 13301 | . . . 4 ⊢ ((𝐵 ⊆ ℝ* ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ sup(𝐵, ℝ*, < )) | |
5 | 1, 3, 4 | syl2anc 583 | . . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) ∧ 𝑥 ∈ 𝐴) → 𝑥 ≤ sup(𝐵, ℝ*, < )) |
6 | 5 | ralrimiva 3138 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ*, < )) |
7 | sstr 3983 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → 𝐴 ⊆ ℝ*) | |
8 | supxrcl 13292 | . . . 4 ⊢ (𝐵 ⊆ ℝ* → sup(𝐵, ℝ*, < ) ∈ ℝ*) | |
9 | 8 | adantl 481 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → sup(𝐵, ℝ*, < ) ∈ ℝ*) |
10 | supxrleub 13303 | . . 3 ⊢ ((𝐴 ⊆ ℝ* ∧ sup(𝐵, ℝ*, < ) ∈ ℝ*) → (sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ*, < ))) | |
11 | 7, 9, 10 | syl2anc 583 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → (sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < ) ↔ ∀𝑥 ∈ 𝐴 𝑥 ≤ sup(𝐵, ℝ*, < ))) |
12 | 6, 11 | mpbird 257 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ*) → sup(𝐴, ℝ*, < ) ≤ sup(𝐵, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ∀wral 3053 ⊆ wss 3941 class class class wbr 5139 supcsup 9432 ℝ*cxr 11245 < clt 11246 ≤ cle 11247 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 |
This theorem is referenced by: deg1mul3le 25976 ioossioobi 44740 limsupres 44931 supcnvlimsup 44966 liminfval2 44994 liminflelimsuplem 45001 sge0less 45618 sge0reuz 45673 smflimsuplem4 46049 |
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