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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 3200 | . . . . 5 | |
2 | invdif 3179 | . . . . 5 | |
3 | 1, 2 | eqtr4i 2063 | . . . 4 |
4 | un0 3251 | . . . 4 | |
5 | 3, 4 | eqtr4i 2063 | . . 3 |
6 | indi 3184 | . . . 4 | |
7 | disjdif 3296 | . . . . . 6 | |
8 | incom 3129 | . . . . . 6 | |
9 | 7, 8 | eqtr3i 2062 | . . . . 5 |
10 | 9 | uneq2i 3094 | . . . 4 |
11 | 6, 10 | eqtr4i 2063 | . . 3 |
12 | 5, 11 | eqtr4i 2063 | . 2 |
13 | ddifss 3175 | . . . . . 6 | |
14 | unss2 3114 | . . . . . 6 | |
15 | 13, 14 | ax-mp 7 | . . . . 5 |
16 | indmss 3196 | . . . . . 6 | |
17 | invdif 3179 | . . . . . . 7 | |
18 | 17 | difeq2i 3059 | . . . . . 6 |
19 | 16, 18 | sseqtri 2977 | . . . . 5 |
20 | 15, 19 | sstri 2954 | . . . 4 |
21 | sslin 3163 | . . . 4 | |
22 | 20, 21 | ax-mp 7 | . . 3 |
23 | invdif 3179 | . . 3 | |
24 | 22, 23 | sseqtri 2977 | . 2 |
25 | 12, 24 | eqsstri 2975 | 1 |
Colors of variables: wff set class |
Syntax hints: cvv 2557 cdif 2914 cun 2915 cin 2916 wss 2917 c0 3224 |
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-ral 2311 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 |
This theorem is referenced by: (None) |
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