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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 3318 |
. . . . . 6
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2 | 1 | imbi1i 238 |
. . . . 5
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3 | imanim 688 |
. . . . 5
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4 | 2, 3 | sylbi 121 |
. . . 4
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5 | eldif 3138 |
. . . . . 6
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6 | 5 | anbi2i 457 |
. . . . 5
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7 | elin 3318 |
. . . . 5
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8 | anass 401 |
. . . . 5
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9 | 6, 7, 8 | 3bitr4ri 213 |
. . . 4
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10 | 4, 9 | sylnib 676 |
. . 3
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11 | 10 | alimi 1455 |
. 2
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12 | dfss2 3144 |
. 2
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13 | eq0 3441 |
. 2
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14 | 11, 12, 13 | 3imtr4i 201 |
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-in1 614 ax-in2 615 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-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 |
This theorem is referenced by: disjdif 3495 |
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