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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 3390 | . . . . 5 | |
2 | invdif 3369 | . . . . 5 | |
3 | 1, 2 | eqtr4i 2194 | . . . 4 |
4 | un0 3448 | . . . 4 | |
5 | 3, 4 | eqtr4i 2194 | . . 3 |
6 | indi 3374 | . . . 4 | |
7 | disjdif 3487 | . . . . . 6 | |
8 | incom 3319 | . . . . . 6 | |
9 | 7, 8 | eqtr3i 2193 | . . . . 5 |
10 | 9 | uneq2i 3278 | . . . 4 |
11 | 6, 10 | eqtr4i 2194 | . . 3 |
12 | 5, 11 | eqtr4i 2194 | . 2 |
13 | ddifss 3365 | . . . . . 6 | |
14 | unss2 3298 | . . . . . 6 | |
15 | 13, 14 | ax-mp 5 | . . . . 5 |
16 | indmss 3386 | . . . . . 6 | |
17 | invdif 3369 | . . . . . . 7 | |
18 | 17 | difeq2i 3242 | . . . . . 6 |
19 | 16, 18 | sseqtri 3181 | . . . . 5 |
20 | 15, 19 | sstri 3156 | . . . 4 |
21 | sslin 3353 | . . . 4 | |
22 | 20, 21 | ax-mp 5 | . . 3 |
23 | invdif 3369 | . . 3 | |
24 | 22, 23 | sseqtri 3181 | . 2 |
25 | 12, 24 | eqsstri 3179 | 1 |
Colors of variables: wff set class |
Syntax hints: cvv 2730 cdif 3118 cun 3119 cin 3120 wss 3121 c0 3414 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 |
This theorem is referenced by: (None) |
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