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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsigagen2 | Structured version Unicode version |
Description: Any countable union of elements of a set is also in the sigma algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.) |
Ref | Expression |
---|---|
elsigagen2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 988 |
. . 3
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2 | 1 | sgsiga 26753 |
. 2
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3 | sssigagen 26756 |
. . . 4
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4 | sspwb 4652 |
. . . . 5
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5 | 4 | biimpi 194 |
. . . 4
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6 | 1, 3, 5 | 3syl 20 |
. . 3
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7 | simp2 989 |
. . . 4
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8 | simp3 990 |
. . . . 5
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9 | ctex 26186 |
. . . . 5
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10 | elpwg 3979 |
. . . . 5
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11 | 8, 9, 10 | 3syl 20 |
. . . 4
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12 | 7, 11 | mpbird 232 |
. . 3
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13 | 6, 12 | sseldd 3468 |
. 2
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14 | sigaclcu 26728 |
. 2
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15 | 2, 13, 8, 14 | syl3anc 1219 |
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-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 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 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3399 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-int 4240 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-res 4963 df-ima 4964 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-f1 5534 df-fv 5537 df-dom 7425 df-siga 26719 df-sigagen 26750 |
This theorem is referenced by: sxbrsigalem1 26867 |
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