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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 3126 |
. . . . . 6
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2 | 1 | imbi1i 227 |
. . . . 5
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3 | imanim 785 |
. . . . 5
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4 | 2, 3 | sylbi 114 |
. . . 4
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5 | eldif 2927 |
. . . . . 6
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6 | 5 | anbi2i 430 |
. . . . 5
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7 | elin 3126 |
. . . . 5
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8 | anass 381 |
. . . . 5
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9 | 6, 7, 8 | 3bitr4ri 202 |
. . . 4
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10 | 4, 9 | sylnib 601 |
. . 3
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11 | 10 | alimi 1344 |
. 2
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12 | dfss2 2934 |
. 2
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13 | eq0 3239 |
. 2
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14 | 11, 12, 13 | 3imtr4i 190 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-in 2924 df-ss 2931 df-nul 3225 |
This theorem is referenced by: disjdif 3296 |
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