![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfun3 | Structured version Visualization version GIF version |
Description: Union defined in terms of intersection (De Morgan's law). Definition of union in [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
dfun3 | ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfun2 4276 | . 2 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) | |
2 | dfin2 4277 | . . . 4 ⊢ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) = ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) | |
3 | ddif 4151 | . . . . 5 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
4 | 3 | difeq2i 4133 | . . . 4 ⊢ ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) = ((V ∖ 𝐴) ∖ 𝐵) |
5 | 2, 4 | eqtr2i 2764 | . . 3 ⊢ ((V ∖ 𝐴) ∖ 𝐵) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) |
6 | 5 | difeq2i 4133 | . 2 ⊢ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
7 | 1, 6 | eqtri 2763 | 1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 |
This theorem is referenced by: difundi 4296 unvdif 4481 |
Copyright terms: Public domain | W3C validator |