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Mirrors > Home > MPE Home > Th. List > rpsup | Structured version Visualization version GIF version |
Description: The positive reals are unbounded above. (Contributed by Mario Carneiro, 7-May-2016.) |
Ref | Expression |
---|---|
rpsup | ⊢ sup(ℝ+, ℝ*, < ) = +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioorp 12804 | . . 3 ⊢ (0(,)+∞) = ℝ+ | |
2 | 1 | supeq1i 8900 | . 2 ⊢ sup((0(,)+∞), ℝ*, < ) = sup(ℝ+, ℝ*, < ) |
3 | 0xr 10677 | . . 3 ⊢ 0 ∈ ℝ* | |
4 | 0re 10632 | . . . 4 ⊢ 0 ∈ ℝ | |
5 | renepnf 10678 | . . . 4 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 0 ≠ +∞ |
7 | ioopnfsup 13222 | . . 3 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ +∞) → sup((0(,)+∞), ℝ*, < ) = +∞) | |
8 | 3, 6, 7 | mp2an 688 | . 2 ⊢ sup((0(,)+∞), ℝ*, < ) = +∞ |
9 | 2, 8 | eqtr3i 2846 | 1 ⊢ sup(ℝ+, ℝ*, < ) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ≠ wne 3016 (class class class)co 7145 supcsup 8893 ℝcr 10525 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ℝ+crp 12379 (,)cioo 12728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 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 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-sup 8895 df-inf 8896 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-rp 12380 df-ioo 12732 |
This theorem is referenced by: divsqrtsumo1 25489 dchrisum0lem3 26023 mulog2sumlem1 26038 |
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