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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > carsguni | Structured version Visualization version GIF version |
Description: The union of all Caratheodory measurable sets is the universe. (Contributed by Thierry Arnoux, 22-May-2020.) |
Ref | Expression |
---|---|
carsgval.1 | β’ (π β π β π) |
carsgval.2 | β’ (π β π:π« πβΆ(0[,]+β)) |
baselcarsg.1 | β’ (π β (πββ ) = 0) |
Ref | Expression |
---|---|
carsguni | β’ (π β βͺ (toCaraSigaβπ) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carsgval.1 | . . . . . . 7 β’ (π β π β π) | |
2 | carsgval.2 | . . . . . . 7 β’ (π β π:π« πβΆ(0[,]+β)) | |
3 | 1, 2 | carsgcl 33298 | . . . . . 6 β’ (π β (toCaraSigaβπ) β π« π) |
4 | 3 | sselda 3982 | . . . . 5 β’ ((π β§ π β (toCaraSigaβπ)) β π β π« π) |
5 | 4 | elpwid 4611 | . . . 4 β’ ((π β§ π β (toCaraSigaβπ)) β π β π) |
6 | 5 | ralrimiva 3146 | . . 3 β’ (π β βπ β (toCaraSigaβπ)π β π) |
7 | unissb 4943 | . . 3 β’ (βͺ (toCaraSigaβπ) β π β βπ β (toCaraSigaβπ)π β π) | |
8 | 6, 7 | sylibr 233 | . 2 β’ (π β βͺ (toCaraSigaβπ) β π) |
9 | baselcarsg.1 | . . 3 β’ (π β (πββ ) = 0) | |
10 | 1, 2, 9 | baselcarsg 33300 | . 2 β’ (π β π β (toCaraSigaβπ)) |
11 | unissel 4942 | . 2 β’ ((βͺ (toCaraSigaβπ) β π β§ π β (toCaraSigaβπ)) β βͺ (toCaraSigaβπ) = π) | |
12 | 8, 10, 11 | syl2anc 584 | 1 β’ (π β βͺ (toCaraSigaβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β wss 3948 β c0 4322 π« cpw 4602 βͺ cuni 4908 βΆwf 6539 βcfv 6543 (class class class)co 7408 0cc0 11109 +βcpnf 11244 [,]cicc 13326 toCaraSigaccarsg 33295 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-xadd 13092 df-icc 13330 df-carsg 33296 |
This theorem is referenced by: carsgclctun 33315 |
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