Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salincl | Structured version Visualization version Unicode version |
Description: The intersection of two sets in a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
salincl | SAlg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2623 | . . 3 SAlg | |
2 | inss1 3833 | . . . . . . . 8 | |
3 | 2 | a1i 11 | . . . . . . 7 SAlg |
4 | elssuni 4467 | . . . . . . . 8 | |
5 | 4 | adantl 482 | . . . . . . 7 SAlg |
6 | 3, 5 | sstrd 3613 | . . . . . 6 SAlg |
7 | dfss4 3858 | . . . . . 6 | |
8 | 6, 7 | sylib 208 | . . . . 5 SAlg |
9 | 8 | eqcomd 2628 | . . . 4 SAlg |
10 | 9 | 3adant3 1081 | . . 3 SAlg |
11 | difindi 3881 | . . . . 5 | |
12 | 11 | difeq2i 3725 | . . . 4 |
13 | 12 | a1i 11 | . . 3 SAlg |
14 | 1, 10, 13 | 3eqtrd 2660 | . 2 SAlg |
15 | simp1 1061 | . . 3 SAlg SAlg | |
16 | saldifcl 40539 | . . . . 5 SAlg | |
17 | 16 | 3adant3 1081 | . . . 4 SAlg |
18 | saldifcl 40539 | . . . . 5 SAlg | |
19 | 18 | 3adant2 1080 | . . . 4 SAlg |
20 | saluncl 40537 | . . . 4 SAlg | |
21 | 15, 17, 19, 20 | syl3anc 1326 | . . 3 SAlg |
22 | saldifcl 40539 | . . 3 SAlg | |
23 | 15, 21, 22 | syl2anc 693 | . 2 SAlg |
24 | 14, 23 | eqeltrd 2701 | 1 SAlg |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cdif 3571 cun 3572 cin 3573 wss 3574 cuni 4436 SAlgcsalg 40528 |
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 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-salg 40529 |
This theorem is referenced by: saldifcl2 40546 salincld 40570 |
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