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Mirrors > Home > ILE Home > Th. List > disjxsn | Unicode version |
Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
disjxsn | Disj |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisj2 3966 | . 2 Disj | |
2 | moeq 2905 | . . 3 | |
3 | elsni 3599 | . . . . 5 | |
4 | 3 | adantr 274 | . . . 4 |
5 | 4 | moimi 2084 | . . 3 |
6 | 2, 5 | ax-mp 5 | . 2 |
7 | 1, 6 | mpgbir 1446 | 1 Disj |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wmo 2020 wcel 2141 csn 3581 Disj wdisj 3964 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rmo 2456 df-v 2732 df-sn 3587 df-disj 3965 |
This theorem is referenced by: disjx0 3986 |
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