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Mirrors > Home > MPE Home > Th. List > dfun2 | Structured version Visualization version GIF version |
Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 4161 and dfss4 4159 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation ∖ (class difference). (Contributed by NM, 10-Jun-2004.) |
Ref | Expression |
---|---|
dfun2 | ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3402 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | eldif 3863 | . . . . . . 7 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | mpbiran 709 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) |
4 | 3 | anbi1i 627 | . . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
5 | eldif 3863 | . . . . 5 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵)) | |
6 | ioran 984 | . . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
7 | 4, 5, 6 | 3bitr4i 306 | . . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) |
8 | 7 | con2bii 361 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)) |
9 | eldif 3863 | . . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))) | |
10 | 1, 9 | mpbiran 709 | . . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)) |
11 | 8, 10 | bitr4i 281 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))) |
12 | 11 | uneqri 4051 | 1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∖ cdif 3850 ∪ cun 3851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-dif 3856 df-un 3858 |
This theorem is referenced by: dfun3 4166 dfin3 4167 |
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