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Mirrors > Home > NFE Home > Th. List > difdifdir | Unicode version |
Description: Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.) |
Ref | Expression |
---|---|
difdifdir |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif32 3517 | . . . . 5 | |
2 | invdif 3496 | . . . . 5 | |
3 | 1, 2 | eqtr4i 2376 | . . . 4 |
4 | un0 3575 | . . . 4 | |
5 | 3, 4 | eqtr4i 2376 | . . 3 |
6 | indi 3501 | . . . 4 | |
7 | disjdif 3622 | . . . . . 6 | |
8 | incom 3448 | . . . . . 6 | |
9 | 7, 8 | eqtr3i 2375 | . . . . 5 |
10 | 9 | uneq2i 3415 | . . . 4 |
11 | 6, 10 | eqtr4i 2376 | . . 3 |
12 | 5, 11 | eqtr4i 2376 | . 2 |
13 | ddif 3398 | . . . . 5 | |
14 | 13 | uneq2i 3415 | . . . 4 |
15 | indm 3513 | . . . . 5 | |
16 | invdif 3496 | . . . . . 6 | |
17 | 16 | difeq2i 3382 | . . . . 5 |
18 | 15, 17 | eqtr3i 2375 | . . . 4 |
19 | 14, 18 | eqtr3i 2375 | . . 3 |
20 | 19 | ineq2i 3454 | . 2 |
21 | invdif 3496 | . 2 | |
22 | 12, 20, 21 | 3eqtri 2377 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1642 cvv 2859 cdif 3206 cun 3207 cin 3208 c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: (None) |
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