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Mirrors > Home > NFE Home > Th. List > iinun | GIF version |
Description: Complement of intersection is equal to union of complements. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
iinun | ⊢ ∼ (A ∩ B) = ( ∼ A ∪ ∼ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfun4 3546 | . 2 ⊢ ( ∼ A ∪ ∼ B) = ∼ ( ∼ ∼ A ∩ ∼ ∼ B) | |
2 | dblcompl 3227 | . . . 4 ⊢ ∼ ∼ A = A | |
3 | dblcompl 3227 | . . . 4 ⊢ ∼ ∼ B = B | |
4 | 2, 3 | ineq12i 3455 | . . 3 ⊢ ( ∼ ∼ A ∩ ∼ ∼ B) = (A ∩ B) |
5 | 4 | compleqi 3244 | . 2 ⊢ ∼ ( ∼ ∼ A ∩ ∼ ∼ B) = ∼ (A ∩ B) |
6 | 1, 5 | eqtr2i 2374 | 1 ⊢ ∼ (A ∩ B) = ( ∼ A ∪ ∼ B) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∼ ccompl 3205 ∪ cun 3207 ∩ cin 3208 |
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-un 3214 |
This theorem is referenced by: sbthlem1 6203 |
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