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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 3122 | . . . . 5 | |
2 | 1 | ralimi 2520 | . . . 4 |
3 | rmoim 2913 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | 4 | alimdv 1859 | . 2 |
6 | df-disj 3945 | . 2 Disj | |
7 | df-disj 3945 | . 2 Disj | |
8 | 5, 6, 7 | 3imtr4g 204 | 1 Disj Disj |
Colors of variables: wff set class |
Syntax hints: wi 4 wal 1333 wcel 2128 wral 2435 wrmo 2438 wss 3102 Disj wdisj 3944 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-ral 2440 df-rmo 2443 df-in 3108 df-ss 3115 df-disj 3945 |
This theorem is referenced by: disjeq2 3948 0disj 3964 |
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