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Mirrors > Home > ILE Home > Th. List > inssdif0im | Unicode version |
Description: Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.) |
Ref | Expression |
---|---|
inssdif0im |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3264 |
. . . . . 6
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2 | 1 | imbi1i 237 |
. . . . 5
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3 | imanim 678 |
. . . . 5
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4 | 2, 3 | sylbi 120 |
. . . 4
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5 | eldif 3085 |
. . . . . 6
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6 | 5 | anbi2i 453 |
. . . . 5
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7 | elin 3264 |
. . . . 5
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8 | anass 399 |
. . . . 5
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9 | 6, 7, 8 | 3bitr4ri 212 |
. . . 4
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10 | 4, 9 | sylnib 666 |
. . 3
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11 | 10 | alimi 1432 |
. 2
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12 | dfss2 3091 |
. 2
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13 | eq0 3386 |
. 2
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14 | 11, 12, 13 | 3imtr4i 200 |
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-in 3082 df-ss 3089 df-nul 3369 |
This theorem is referenced by: disjdif 3440 |
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