![]() |
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 4224 | . 2 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) | |
2 | dfin2 4225 | . . . 4 ⊢ ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) = ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) | |
3 | ddif 4101 | . . . . 5 ⊢ (V ∖ (V ∖ 𝐵)) = 𝐵 | |
4 | 3 | difeq2i 4084 | . . . 4 ⊢ ((V ∖ 𝐴) ∖ (V ∖ (V ∖ 𝐵))) = ((V ∖ 𝐴) ∖ 𝐵) |
5 | 2, 4 | eqtr2i 2766 | . . 3 ⊢ ((V ∖ 𝐴) ∖ 𝐵) = ((V ∖ 𝐴) ∩ (V ∖ 𝐵)) |
6 | 5 | difeq2i 4084 | . 2 ⊢ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
7 | 1, 6 | eqtri 2765 | 1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3448 ∖ cdif 3912 ∪ cun 3913 ∩ cin 3914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 |
This theorem is referenced by: difundi 4244 unvdif 4439 |
Copyright terms: Public domain | W3C validator |