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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3164 |
. . . . 5
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2 | 1 | ralimi 2553 |
. . . 4
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3 | rmoim 2953 |
. . . 4
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4 | 2, 3 | syl 14 |
. . 3
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5 | 4 | alimdv 1890 |
. 2
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6 | df-disj 3996 |
. 2
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7 | df-disj 3996 |
. 2
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8 | 5, 6, 7 | 3imtr4g 205 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-ral 2473 df-rmo 2476 df-in 3150 df-ss 3157 df-disj 3996 |
This theorem is referenced by: disjeq2 3999 0disj 4015 |
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