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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 3978 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
2 | supxrcl 13294 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
3 | 2 | adantr 482 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
4 | xrltso 13120 | . . . . 5 ⊢ < Or ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
6 | xrsupss 13288 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
7 | 5, 6 | supub 9454 | . . 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 11282 | 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 3949 class class class wbr 5149 Or wor 5588 supcsup 9435 ℝ*cxr 11247 < clt 11248 ≤ cle 11249 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 |
This theorem is referenced by: supxrre 13306 supxrss 13311 ixxub 13345 prdsdsf 23873 prdsxmetlem 23874 xpsdsval 23887 prdsbl 24000 xrge0tsms 24350 bndth 24474 ovolmge0 24994 ovollb2lem 25005 ovolunlem1a 25013 ovoliunlem1 25019 ovoliun 25022 ovolicc2lem4 25037 ioombl1lem2 25076 ioombl1lem4 25078 uniioombllem2 25100 uniioombllem3 25102 uniioombllem6 25105 vitalilem4 25128 itg2ub 25251 itg2seq 25260 itg2monolem1 25268 itg2monolem2 25269 itg2monolem3 25270 aannenlem2 25842 radcnvcl 25929 radcnvle 25932 nmooge0 30020 nmoolb 30024 nmlno0lem 30046 nmoplb 31160 nmfnlb 31177 nmlnop0iALT 31248 xrofsup 31980 xrge0tsmsd 32209 itg2addnc 36542 rrnequiv 36703 supxrubd 43802 supxrgere 44043 supxrgelem 44047 suplesup2 44086 ressiocsup 44267 ressioosup 44268 liminfval2 44484 etransclem48 44998 fsumlesge0 45093 sge0cl 45097 sge0supre 45105 sge0xaddlem1 45149 sge0xaddlem2 45150 |
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