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Mirrors > Home > MPE Home > Th. List > ovolge0 | Structured version Visualization version Unicode version |
Description: The volume of a set is always nonnegative. (Contributed by Mario Carneiro, 16-Mar-2014.) |
Ref | Expression |
---|---|
ovolge0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3516 |
. . . 4
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2 | 0xr 9692 |
. . . 4
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3 | infxrgelb 11628 |
. . . 4
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4 | 1, 2, 3 | mp2an 679 |
. . 3
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5 | eqid 2453 |
. . . 4
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6 | 5 | ovolmge0 22442 |
. . 3
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7 | 4, 6 | mprgbir 2754 |
. 2
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8 | 5 | ovolval 22438 |
. 2
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9 | 7, 8 | syl5breqr 4442 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-8 1891 ax-9 1898 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 ax-sep 4528 ax-nul 4537 ax-pow 4584 ax-pr 4642 ax-un 6588 ax-cnex 9600 ax-resscn 9601 ax-1cn 9602 ax-icn 9603 ax-addcl 9604 ax-addrcl 9605 ax-mulcl 9606 ax-mulrcl 9607 ax-mulcom 9608 ax-addass 9609 ax-mulass 9610 ax-distr 9611 ax-i2m1 9612 ax-1ne0 9613 ax-1rid 9614 ax-rnegex 9615 ax-rrecex 9616 ax-cnre 9617 ax-pre-lttri 9618 ax-pre-lttrn 9619 ax-pre-ltadd 9620 ax-pre-mulgt0 9621 ax-pre-sup 9622 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 987 df-3an 988 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2583 df-ne 2626 df-nel 2627 df-ral 2744 df-rex 2745 df-reu 2746 df-rmo 2747 df-rab 2748 df-v 3049 df-sbc 3270 df-csb 3366 df-dif 3409 df-un 3411 df-in 3413 df-ss 3420 df-pss 3422 df-nul 3734 df-if 3884 df-pw 3955 df-sn 3971 df-pr 3973 df-tp 3975 df-op 3977 df-uni 4202 df-iun 4283 df-br 4406 df-opab 4465 df-mpt 4466 df-tr 4501 df-eprel 4748 df-id 4752 df-po 4758 df-so 4759 df-fr 4796 df-we 4798 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-pred 5383 df-ord 5429 df-on 5430 df-lim 5431 df-suc 5432 df-iota 5549 df-fun 5587 df-fn 5588 df-f 5589 df-f1 5590 df-fo 5591 df-f1o 5592 df-fv 5593 df-riota 6257 df-ov 6298 df-oprab 6299 df-mpt2 6300 df-om 6698 df-1st 6798 df-2nd 6799 df-wrecs 7033 df-recs 7095 df-rdg 7133 df-er 7368 df-map 7479 df-en 7575 df-dom 7576 df-sdom 7577 df-sup 7961 df-inf 7962 df-pnf 9682 df-mnf 9683 df-xr 9684 df-ltxr 9685 df-le 9686 df-sub 9867 df-neg 9868 df-div 10277 df-nn 10617 df-2 10675 df-3 10676 df-n0 10877 df-z 10945 df-uz 11167 df-rp 11310 df-ico 11648 df-fz 11792 df-seq 12221 df-exp 12280 df-cj 13174 df-re 13175 df-im 13176 df-sqrt 13310 df-abs 13311 df-ovol 22428 |
This theorem is referenced by: ovolf 22447 ovollecl 22448 ovolssnul 22452 ovolctb 22455 ovolunnul 22465 ovoliun2 22471 ovoliunnul 22472 ovolicopnf 22490 voliunlem3 22517 volsup 22521 ioorcl2 22536 vitalilem4 22581 vitalilem5 22582 itg1ge0 22656 itg1ge0a 22681 itgulm 23375 areaf 23899 mblfinlem3 31991 mblfinlem4 31992 ismblfin 31993 ovoliunnfl 31994 volsupnfl 31997 itg2addnclem 32005 volge0 37848 |
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