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Mirrors > Home > ILE Home > Th. List > disjss1 | Unicode version |
Description: A subset of a disjoint collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjss1 | Disj Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3086 | . . . . . 6 | |
2 | 1 | anim1d 334 | . . . . 5 |
3 | 2 | alrimiv 1846 | . . . 4 |
4 | moim 2061 | . . . 4 | |
5 | 3, 4 | syl 14 | . . 3 |
6 | 5 | alimdv 1851 | . 2 |
7 | dfdisj2 3903 | . 2 Disj | |
8 | dfdisj2 3903 | . 2 Disj | |
9 | 6, 7, 8 | 3imtr4g 204 | 1 Disj Disj |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1329 wcel 1480 wmo 1998 wss 3066 Disj wdisj 3901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-rmo 2422 df-in 3072 df-ss 3079 df-disj 3902 |
This theorem is referenced by: disjeq1 3908 disjx0 3923 fsumiun 11239 |
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