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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif32 3309 | . . . . 5 | |
2 | invdif 3288 | . . . . 5 | |
3 | 1, 2 | eqtr4i 2141 | . . . 4 |
4 | un0 3366 | . . . 4 | |
5 | 3, 4 | eqtr4i 2141 | . . 3 |
6 | indi 3293 | . . . 4 | |
7 | disjdif 3405 | . . . . . 6 | |
8 | incom 3238 | . . . . . 6 | |
9 | 7, 8 | eqtr3i 2140 | . . . . 5 |
10 | 9 | uneq2i 3197 | . . . 4 |
11 | 6, 10 | eqtr4i 2141 | . . 3 |
12 | 5, 11 | eqtr4i 2141 | . 2 |
13 | ddifss 3284 | . . . . . 6 | |
14 | unss2 3217 | . . . . . 6 | |
15 | 13, 14 | ax-mp 5 | . . . . 5 |
16 | indmss 3305 | . . . . . 6 | |
17 | invdif 3288 | . . . . . . 7 | |
18 | 17 | difeq2i 3161 | . . . . . 6 |
19 | 16, 18 | sseqtri 3101 | . . . . 5 |
20 | 15, 19 | sstri 3076 | . . . 4 |
21 | sslin 3272 | . . . 4 | |
22 | 20, 21 | ax-mp 5 | . . 3 |
23 | invdif 3288 | . . 3 | |
24 | 22, 23 | sseqtri 3101 | . 2 |
25 | 12, 24 | eqsstri 3099 | 1 |
Colors of variables: wff set class |
Syntax hints: cvv 2660 cdif 3038 cun 3039 cin 3040 wss 3041 c0 3333 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 |
This theorem is referenced by: (None) |
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