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Mirrors > Home > ILE Home > Th. List > unssin | Unicode version |
Description: Union as a subset of class complement and intersection (De Morgan's law). One direction of the definition of union in [Mendelson] p. 231. This would be an equality, rather than subset, in classical logic. (Contributed by Jim Kingdon, 25-Jul-2018.) |
Ref | Expression |
---|---|
unssin |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oranim 771 |
. . . . 5
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2 | eldifn 3204 |
. . . . . 6
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3 | eldifn 3204 |
. . . . . 6
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4 | 2, 3 | anim12i 336 |
. . . . 5
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5 | 1, 4 | nsyl 618 |
. . . 4
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6 | elin 3264 |
. . . 4
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7 | 5, 6 | sylnibr 667 |
. . 3
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8 | elun 3222 |
. . 3
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9 | vex 2692 |
. . . 4
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10 | eldif 3085 |
. . . 4
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11 | 9, 10 | mpbiran 925 |
. . 3
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12 | 7, 8, 11 | 3imtr4i 200 |
. 2
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13 | 12 | ssriv 3106 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 |
This theorem is referenced by: (None) |
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