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Mirrors > Home > MPE Home > Th. List > dfin3 | Structured version Visualization version GIF version |
Description: Intersection defined in terms of union (De Morgan's law). Similar to Exercise 4.10(n) of [Mendelson] p. 231. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
dfin3 | ⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ddif 4064 | . 2 ⊢ (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵)) | |
2 | dfun2 4186 | . . . 4 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) | |
3 | ddif 4064 | . . . . . 6 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | |
4 | 3 | difeq1i 4046 | . . . . 5 ⊢ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵)) |
5 | 4 | difeq2i 4047 | . . . 4 ⊢ (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵))) |
6 | 2, 5 | eqtri 2821 | . . 3 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵))) |
7 | 6 | difeq2i 4047 | . 2 ⊢ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) |
8 | dfin2 4187 | . 2 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) | |
9 | 1, 7, 8 | 3eqtr4ri 2832 | 1 ⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 |
This theorem is referenced by: difindi 4208 |
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