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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 4115 | . 2 ⊢ (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵)) | |
2 | dfun2 4238 | . . . 4 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) | |
3 | ddif 4115 | . . . . . 6 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | |
4 | 3 | difeq1i 4097 | . . . . 5 ⊢ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵)) |
5 | 4 | difeq2i 4098 | . . . 4 ⊢ (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵))) |
6 | 2, 5 | eqtri 2846 | . . 3 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵))) |
7 | 6 | difeq2i 4098 | . 2 ⊢ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) |
8 | dfin2 4239 | . 2 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) | |
9 | 1, 7, 8 | 3eqtr4ri 2857 | 1 ⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 |
This theorem is referenced by: difindi 4260 |
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