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Mirrors > Home > NFE Home > Th. List > inindif | GIF version |
Description: The intersection of an intersection and a difference is empty. (Contributed by set.mm contributors, 10-Mar-2015.) |
Ref | Expression |
---|---|
inindif | ⊢ ((A ∩ B) ∩ (A ∖ B)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dif 3215 | . . 3 ⊢ (A ∖ B) = (A ∩ ∼ B) | |
2 | 1 | ineq2i 3454 | . 2 ⊢ ((A ∩ B) ∩ (A ∖ B)) = ((A ∩ B) ∩ (A ∩ ∼ B)) |
3 | inindi 3472 | . 2 ⊢ (A ∩ (B ∩ ∼ B)) = ((A ∩ B) ∩ (A ∩ ∼ B)) | |
4 | incompl 4073 | . . . 4 ⊢ (B ∩ ∼ B) = ∅ | |
5 | 4 | ineq2i 3454 | . . 3 ⊢ (A ∩ (B ∩ ∼ B)) = (A ∩ ∅) |
6 | in0 3576 | . . 3 ⊢ (A ∩ ∅) = ∅ | |
7 | 5, 6 | eqtri 2373 | . 2 ⊢ (A ∩ (B ∩ ∼ B)) = ∅ |
8 | 2, 3, 7 | 3eqtr2i 2379 | 1 ⊢ ((A ∩ B) ∩ (A ∖ B)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∼ ccompl 3205 ∖ cdif 3206 ∩ 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-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: sbthlem1 6203 |
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