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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigaclcu | Structured version Visualization version Unicode version |
Description: A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.) |
Ref | Expression |
---|---|
sigaclcu | sigAlgebra |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1062 | . 2 sigAlgebra | |
2 | isrnsiga 30176 | . . . . 5 sigAlgebra | |
3 | 2 | simprbi 480 | . . . 4 sigAlgebra |
4 | simpr3 1069 | . . . . 5 | |
5 | 4 | exlimiv 1858 | . . . 4 |
6 | 3, 5 | syl 17 | . . 3 sigAlgebra |
7 | 6 | 3ad2ant1 1082 | . 2 sigAlgebra |
8 | simp3 1063 | . 2 sigAlgebra | |
9 | breq1 4656 | . . . 4 | |
10 | unieq 4444 | . . . . 5 | |
11 | 10 | eleq1d 2686 | . . . 4 |
12 | 9, 11 | imbi12d 334 | . . 3 |
13 | 12 | rspcv 3305 | . 2 |
14 | 1, 7, 8, 13 | syl3c 66 | 1 sigAlgebra |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wex 1704 wcel 1990 wral 2912 cvv 3200 cdif 3571 wss 3574 cpw 4158 cuni 4436 class class class wbr 4653 crn 5115 com 7065 cdom 7953 sigAlgebracsiga 30170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-siga 30171 |
This theorem is referenced by: sigaclcuni 30181 sigaclfu 30182 sigaclcu2 30183 sigainb 30199 elsigagen2 30211 sigaldsys 30222 measinb 30284 measres 30285 measdivcstOLD 30287 measdivcst 30288 imambfm 30324 totprobd 30488 dstrvprob 30533 |
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