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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 33965 | . . . . . 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 4615 | . . . 4 β’ ((π β§ π β (toCaraSigaβπ)) β π β π) |
6 | 5 | ralrimiva 3143 | . . 3 β’ (π β βπ β (toCaraSigaβπ)π β π) |
7 | unissb 4946 | . . 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 33967 | . 2 β’ (π β π β (toCaraSigaβπ)) |
11 | unissel 4945 | . 2 β’ ((βͺ (toCaraSigaβπ) β π β§ π β (toCaraSigaβπ)) β βͺ (toCaraSigaβπ) = π) | |
12 | 8, 10, 11 | syl2anc 582 | 1 β’ (π β βͺ (toCaraSigaβπ) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3058 β wss 3949 β c0 4326 π« cpw 4606 βͺ cuni 4912 βΆwf 6549 βcfv 6553 (class class class)co 7426 0cc0 11148 +βcpnf 11285 [,]cicc 13369 toCaraSigaccarsg 33962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-xadd 13135 df-icc 13373 df-carsg 33963 |
This theorem is referenced by: carsgclctun 33982 |
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