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Mirrors > Home > MPE Home > Th. List > dfin2 | Structured version Visualization version GIF version |
Description: An alternate definition of the intersection of two classes in terms of class difference, requiring no dummy variables. See comments under dfun2 3859. Another version is given by dfin4 3867. (Contributed by NM, 10-Jun-2004.) |
Ref | Expression |
---|---|
dfin2 | ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | eldif 3584 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | mpbiran 953 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵) |
4 | 3 | con2bii 347 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵)) |
5 | 4 | anbi2i 730 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵))) |
6 | eldif 3584 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵))) | |
7 | 5, 6 | bitr4i 267 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵))) |
8 | 7 | ineqri 3806 | 1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 |
This theorem is referenced by: dfun3 3865 dfin3 3866 invdif 3868 difundi 3879 difindi 3881 difdif2 3884 |
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