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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 12500 | . . 3 ⊢ (0(,)+∞) = ℝ+ | |
2 | 1 | supeq1i 8595 | . 2 ⊢ sup((0(,)+∞), ℝ*, < ) = sup(ℝ+, ℝ*, < ) |
3 | 0xr 10375 | . . 3 ⊢ 0 ∈ ℝ* | |
4 | 0re 10330 | . . . 4 ⊢ 0 ∈ ℝ | |
5 | renepnf 10376 | . . . 4 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 0 ≠ +∞ |
7 | ioopnfsup 12918 | . . 3 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ +∞) → sup((0(,)+∞), ℝ*, < ) = +∞) | |
8 | 3, 6, 7 | mp2an 684 | . 2 ⊢ sup((0(,)+∞), ℝ*, < ) = +∞ |
9 | 2, 8 | eqtr3i 2823 | 1 ⊢ sup(ℝ+, ℝ*, < ) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 ≠ wne 2971 (class class class)co 6878 supcsup 8588 ℝcr 10223 0cc0 10224 +∞cpnf 10360 ℝ*cxr 10362 < clt 10363 ℝ+crp 12074 (,)cioo 12424 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-inf 8591 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-q 12034 df-rp 12075 df-ioo 12428 |
This theorem is referenced by: divsqrtsumo1 25062 dchrisum0lem3 25560 mulog2sumlem1 25575 |
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