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Mirrors > Home > MPE Home > Th. List > opnmbl | Structured version Unicode version |
Description: All open sets are measurable. This proof, via dyadmbl 21212 and uniioombl 21201, shows that it is possible to avoid choice for measurability of open sets and hence continuous functions, which extends the choice-free consequences of Lebesgue measure considerably farther than would otherwise be possible. (Contributed by Mario Carneiro, 26-Mar-2015.) |
Ref | Expression |
---|---|
opnmbl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6206 |
. . . 4
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2 | oveq1 6206 |
. . . . 5
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3 | 2 | oveq1d 6214 |
. . . 4
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4 | 1, 3 | opeq12d 4174 |
. . 3
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5 | oveq2 6207 |
. . . . 5
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6 | 5 | oveq2d 6215 |
. . . 4
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7 | 5 | oveq2d 6215 |
. . . 4
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8 | 6, 7 | opeq12d 4174 |
. . 3
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9 | 4, 8 | cbvmpt2v 6274 |
. 2
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10 | 9 | opnmbllem 21213 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4510 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-inf2 7957 ax-cnex 9448 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 ax-pre-mulgt0 9469 ax-pre-sup 9470 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-fal 1376 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-reu 2805 df-rmo 2806 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-pss 3451 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-tp 3989 df-op 3991 df-uni 4199 df-int 4236 df-iun 4280 df-disj 4370 df-br 4400 df-opab 4458 df-mpt 4459 df-tr 4493 df-eprel 4739 df-id 4743 df-po 4748 df-so 4749 df-fr 4786 df-se 4787 df-we 4788 df-ord 4829 df-on 4830 df-lim 4831 df-suc 4832 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-isom 5534 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-of 6429 df-om 6586 df-1st 6686 df-2nd 6687 df-recs 6941 df-rdg 6975 df-1o 7029 df-2o 7030 df-oadd 7033 df-omul 7034 df-er 7210 df-map 7325 df-pm 7326 df-en 7420 df-dom 7421 df-sdom 7422 df-fin 7423 df-fi 7771 df-sup 7801 df-oi 7834 df-card 8219 df-acn 8222 df-cda 8447 df-pnf 9530 df-mnf 9531 df-xr 9532 df-ltxr 9533 df-le 9534 df-sub 9707 df-neg 9708 df-div 10104 df-nn 10433 df-2 10490 df-3 10491 df-4 10492 df-n0 10690 df-z 10757 df-uz 10972 df-q 11064 df-rp 11102 df-xneg 11199 df-xadd 11200 df-xmul 11201 df-ioo 11414 df-ico 11416 df-icc 11417 df-fz 11554 df-fzo 11665 df-fl 11758 df-seq 11923 df-exp 11982 df-hash 12220 df-cj 12705 df-re 12706 df-im 12707 df-sqr 12841 df-abs 12842 df-clim 13083 df-rlim 13084 df-sum 13281 df-rest 14479 df-topgen 14500 df-psmet 17933 df-xmet 17934 df-met 17935 df-bl 17936 df-mopn 17937 df-top 18634 df-bases 18636 df-topon 18637 df-cmp 19121 df-ovol 21079 df-vol 21080 |
This theorem is referenced by: subopnmbl 21216 mblfinlem3 28577 mblfinlem4 28578 ismblfin 28579 |
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