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Mirrors > Home > ILE Home > Th. List > difdifdirss | Unicode version |
Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.) |
Ref | Expression |
---|---|
difdifdirss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif32 3410 |
. . . . 5
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2 | invdif 3389 |
. . . . 5
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3 | 1, 2 | eqtr4i 2211 |
. . . 4
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4 | un0 3468 |
. . . 4
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5 | 3, 4 | eqtr4i 2211 |
. . 3
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6 | indi 3394 |
. . . 4
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7 | disjdif 3507 |
. . . . . 6
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8 | incom 3339 |
. . . . . 6
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9 | 7, 8 | eqtr3i 2210 |
. . . . 5
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10 | 9 | uneq2i 3298 |
. . . 4
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11 | 6, 10 | eqtr4i 2211 |
. . 3
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12 | 5, 11 | eqtr4i 2211 |
. 2
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13 | ddifss 3385 |
. . . . . 6
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14 | unss2 3318 |
. . . . . 6
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15 | 13, 14 | ax-mp 5 |
. . . . 5
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16 | indmss 3406 |
. . . . . 6
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17 | invdif 3389 |
. . . . . . 7
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18 | 17 | difeq2i 3262 |
. . . . . 6
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19 | 16, 18 | sseqtri 3201 |
. . . . 5
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20 | 15, 19 | sstri 3176 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | sslin 3373 |
. . . 4
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22 | 20, 21 | ax-mp 5 |
. . 3
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23 | invdif 3389 |
. . 3
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24 | 22, 23 | sseqtri 3201 |
. 2
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25 | 12, 24 | eqsstri 3199 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 |
This theorem is referenced by: (None) |
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