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Mirrors > Home > MPE Home > Th. List > supxrub | Structured version Visualization version GIF version |
Description: A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006.) |
Ref | Expression |
---|---|
supxrub | ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 3975 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
2 | supxrcl 13289 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
3 | 2 | adantr 482 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
4 | xrltso 13115 | . . . . 5 ⊢ < Or ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
6 | xrsupss 13283 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
7 | 5, 6 | supub 9449 | . . 3 ⊢ (𝐴 ⊆ ℝ* → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ*, < ) < 𝐵)) |
8 | 7 | imp 408 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → ¬ sup(𝐴, ℝ*, < ) < 𝐵) |
9 | 1, 3, 8 | xrnltled 11277 | 1 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3946 class class class wbr 5146 Or wor 5585 supcsup 9430 ℝ*cxr 11242 < clt 11243 ≤ cle 11244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 |
This theorem is referenced by: supxrre 13301 supxrss 13306 ixxub 13340 prdsdsf 23854 prdsxmetlem 23855 xpsdsval 23868 prdsbl 23981 xrge0tsms 24331 bndth 24455 ovolmge0 24975 ovollb2lem 24986 ovolunlem1a 24994 ovoliunlem1 25000 ovoliun 25003 ovolicc2lem4 25018 ioombl1lem2 25057 ioombl1lem4 25059 uniioombllem2 25081 uniioombllem3 25083 uniioombllem6 25086 vitalilem4 25109 itg2ub 25232 itg2seq 25241 itg2monolem1 25249 itg2monolem2 25250 itg2monolem3 25251 aannenlem2 25823 radcnvcl 25910 radcnvle 25913 nmooge0 29997 nmoolb 30001 nmlno0lem 30023 nmoplb 31137 nmfnlb 31154 nmlnop0iALT 31225 xrofsup 31957 xrge0tsmsd 32186 itg2addnc 36479 rrnequiv 36640 supxrubd 43734 supxrgere 43977 supxrgelem 43981 suplesup2 44020 ressiocsup 44201 ressioosup 44202 liminfval2 44418 etransclem48 44932 fsumlesge0 45027 sge0cl 45031 sge0supre 45039 sge0xaddlem1 45083 sge0xaddlem2 45084 |
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