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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 4141 | . 2 ⊢ (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) = (𝐴 ∖ (V ∖ 𝐵)) | |
| 2 | dfun2 4270 | . . . 4 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) | |
| 3 | ddif 4141 | . . . . . 6 ⊢ (V ∖ (V ∖ 𝐴)) = 𝐴 | |
| 4 | 3 | difeq1i 4122 | . . . . 5 ⊢ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵)) = (𝐴 ∖ (V ∖ 𝐵)) |
| 5 | 4 | difeq2i 4123 | . . . 4 ⊢ (V ∖ ((V ∖ (V ∖ 𝐴)) ∖ (V ∖ 𝐵))) = (V ∖ (𝐴 ∖ (V ∖ 𝐵))) |
| 6 | 2, 5 | eqtri 2765 | . . 3 ⊢ ((V ∖ 𝐴) ∪ (V ∖ 𝐵)) = (V ∖ (𝐴 ∖ (V ∖ 𝐵))) |
| 7 | 6 | difeq2i 4123 | . 2 ⊢ (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) = (V ∖ (V ∖ (𝐴 ∖ (V ∖ 𝐵)))) |
| 8 | dfin2 4271 | . 2 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) | |
| 9 | 1, 7, 8 | 3eqtr4ri 2776 | 1 ⊢ (𝐴 ∩ 𝐵) = (V ∖ ((V ∖ 𝐴) ∪ (V ∖ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 |
| This theorem is referenced by: difindi 4292 |
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