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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 4186. Another version is given by dfin4 4194. (Contributed by NM, 10-Jun-2004.) |
Ref | Expression |
---|---|
dfin2 | ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3444 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | eldif 3891 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | mpbiran 708 | . . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵) |
4 | 3 | con2bii 361 | . . . 4 ⊢ (𝑥 ∈ 𝐵 ↔ ¬ 𝑥 ∈ (V ∖ 𝐵)) |
5 | 4 | anbi2i 625 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵))) |
6 | eldif 3891 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (V ∖ 𝐵))) | |
7 | 5, 6 | bitr4i 281 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐴 ∖ (V ∖ 𝐵))) |
8 | 7 | ineqri 4130 | 1 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∖ (V ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ∩ 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-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-dif 3884 df-in 3888 |
This theorem is referenced by: dfun3 4192 dfin3 4193 invdif 4195 difundi 4206 difindi 4208 difdif2 4211 |
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