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Mirrors > Home > ILE Home > Th. List > sndisj | Unicode version |
Description: Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
sndisj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisj2 3979 |
. 2
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2 | moeq 2912 |
. . 3
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3 | simpr 110 |
. . . . . 6
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4 | velsn 3608 |
. . . . . 6
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5 | 3, 4 | sylib 122 |
. . . . 5
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6 | 5 | eqcomd 2183 |
. . . 4
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7 | 6 | moimi 2091 |
. . 3
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8 | 2, 7 | ax-mp 5 |
. 2
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9 | 1, 8 | mpgbir 1453 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rmo 2463 df-v 2739 df-sn 3597 df-disj 3978 |
This theorem is referenced by: 0disj 3997 disjsnxp 6232 |
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