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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 3910 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
2 | supxrcl 12696 | . . 3 ⊢ (𝐴 ⊆ ℝ* → sup(𝐴, ℝ*, < ) ∈ ℝ*) | |
3 | 2 | adantr 484 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → sup(𝐴, ℝ*, < ) ∈ ℝ*) |
4 | xrltso 12522 | . . . . 5 ⊢ < Or ℝ* | |
5 | 4 | a1i 11 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → < Or ℝ*) |
6 | xrsupss 12690 | . . . 4 ⊢ (𝐴 ⊆ ℝ* → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) | |
7 | 5, 6 | supub 8907 | . . 3 ⊢ (𝐴 ⊆ ℝ* → (𝐵 ∈ 𝐴 → ¬ sup(𝐴, ℝ*, < ) < 𝐵)) |
8 | 7 | imp 410 | . 2 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → ¬ sup(𝐴, ℝ*, < ) < 𝐵) |
9 | 1, 3, 8 | xrnltled 10698 | 1 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴) → 𝐵 ≤ sup(𝐴, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 Or wor 5437 supcsup 8888 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 |
This theorem is referenced by: supxrre 12708 supxrss 12713 ixxub 12747 prdsdsf 22974 prdsxmetlem 22975 xpsdsval 22988 prdsbl 23098 xrge0tsms 23439 bndth 23563 ovolmge0 24081 ovollb2lem 24092 ovolunlem1a 24100 ovoliunlem1 24106 ovoliun 24109 ovolicc2lem4 24124 ioombl1lem2 24163 ioombl1lem4 24165 uniioombllem2 24187 uniioombllem3 24189 uniioombllem6 24192 vitalilem4 24215 itg2ub 24337 itg2seq 24346 itg2monolem1 24354 itg2monolem2 24355 itg2monolem3 24356 aannenlem2 24925 radcnvcl 25012 radcnvle 25015 nmooge0 28550 nmoolb 28554 nmlno0lem 28576 nmoplb 29690 nmfnlb 29707 nmlnop0iALT 29778 xrofsup 30518 xrge0tsmsd 30742 itg2addnc 35111 rrnequiv 35273 supxrubd 41749 supxrgere 41965 supxrgelem 41969 suplesup2 42008 ressiocsup 42191 ressioosup 42192 liminfval2 42410 etransclem48 42924 fsumlesge0 43016 sge0cl 43020 sge0supre 43028 sge0xaddlem1 43072 sge0xaddlem2 43073 |
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