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Mirrors > Home > ILE Home > Th. List > disjss2 | Unicode version |
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjss2 | Disj Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3091 | . . . . 5 | |
2 | 1 | ralimi 2495 | . . . 4 |
3 | rmoim 2885 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | 4 | alimdv 1851 | . 2 |
6 | df-disj 3907 | . 2 Disj | |
7 | df-disj 3907 | . 2 Disj | |
8 | 5, 6, 7 | 3imtr4g 204 | 1 Disj Disj |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1329 wcel 1480 wral 2416 wrmo 2419 wss 3071 Disj wdisj 3906 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-ral 2421 df-rmo 2424 df-in 3077 df-ss 3084 df-disj 3907 |
This theorem is referenced by: disjeq2 3910 0disj 3926 |
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