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Mirrors > Home > NFE Home > Th. List > difun1 | GIF version |
Description: A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.) |
Ref | Expression |
---|---|
difun1 | ⊢ (A ∖ (B ∪ C)) = ((A ∖ B) ∖ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 3465 | . . . 4 ⊢ ((A ∩ (V ∖ B)) ∩ (V ∖ C)) = (A ∩ ((V ∖ B) ∩ (V ∖ C))) | |
2 | invdif 3496 | . . . 4 ⊢ ((A ∩ (V ∖ B)) ∩ (V ∖ C)) = ((A ∩ (V ∖ B)) ∖ C) | |
3 | 1, 2 | eqtr3i 2375 | . . 3 ⊢ (A ∩ ((V ∖ B) ∩ (V ∖ C))) = ((A ∩ (V ∖ B)) ∖ C) |
4 | undm 3512 | . . . . 5 ⊢ (V ∖ (B ∪ C)) = ((V ∖ B) ∩ (V ∖ C)) | |
5 | 4 | ineq2i 3454 | . . . 4 ⊢ (A ∩ (V ∖ (B ∪ C))) = (A ∩ ((V ∖ B) ∩ (V ∖ C))) |
6 | invdif 3496 | . . . 4 ⊢ (A ∩ (V ∖ (B ∪ C))) = (A ∖ (B ∪ C)) | |
7 | 5, 6 | eqtr3i 2375 | . . 3 ⊢ (A ∩ ((V ∖ B) ∩ (V ∖ C))) = (A ∖ (B ∪ C)) |
8 | 3, 7 | eqtr3i 2375 | . 2 ⊢ ((A ∩ (V ∖ B)) ∖ C) = (A ∖ (B ∪ C)) |
9 | invdif 3496 | . . 3 ⊢ (A ∩ (V ∖ B)) = (A ∖ B) | |
10 | 9 | difeq1i 3381 | . 2 ⊢ ((A ∩ (V ∖ B)) ∖ C) = ((A ∖ B) ∖ C) |
11 | 8, 10 | eqtr3i 2375 | 1 ⊢ (A ∖ (B ∪ C)) = ((A ∖ B) ∖ C) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 Vcvv 2859 ∖ cdif 3206 ∪ cun 3207 ∩ cin 3208 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 |
This theorem is referenced by: dif32 3517 difabs 3518 |
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