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Mirrors > Home > MPE Home > Th. List > disjss1 | Structured version Visualization version Unicode version |
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjss1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3428 |
. . . . . 6
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2 | 1 | anim1d 568 |
. . . . 5
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3 | 2 | alrimiv 1775 |
. . . 4
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4 | moim 2350 |
. . . 4
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5 | 3, 4 | syl 17 |
. . 3
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6 | 5 | alimdv 1765 |
. 2
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7 | dfdisj2 4378 |
. 2
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8 | dfdisj2 4378 |
. 2
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9 | 6, 7, 8 | 3imtr4g 274 |
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 1671 ax-4 1684 ax-5 1760 ax-6 1807 ax-7 1853 ax-10 1917 ax-11 1922 ax-12 1935 ax-13 2093 ax-ext 2433 |
This theorem depends on definitions: df-bi 189 df-an 373 df-tru 1449 df-ex 1666 df-nf 1670 df-sb 1800 df-eu 2305 df-mo 2306 df-clab 2440 df-cleq 2446 df-clel 2449 df-rmo 2747 df-in 3413 df-ss 3420 df-disj 4377 |
This theorem is referenced by: disjeq1 4383 disjx0 4400 disjxiun 4402 disjss3 4404 volfiniun 22512 uniioovol 22548 uniioombllem4 22556 disjiunel 28218 carsggect 29162 carsgclctunlem2 29163 omsmeas 29167 omsmeasOLD 29168 sibfof 29185 disjf1o 37476 fsumiunss 37664 sge0iunmptlemre 38267 meadjiunlem 38313 |
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