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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 4225 and dfss4 4223 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 3452 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | eldif 3925 | . . . . . . 7 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴)) | |
3 | 1, 2 | mpbiran 708 | . . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴) |
4 | 3 | anbi1i 625 | . . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) |
5 | eldif 3925 | . . . . 5 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵)) | |
6 | ioran 983 | . . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵)) | |
7 | 4, 5, 6 | 3bitr4i 303 | . . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) |
8 | 7 | con2bii 358 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)) |
9 | eldif 3925 | . . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))) | |
10 | 1, 9 | mpbiran 708 | . . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)) |
11 | 8, 10 | bitr4i 278 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵))) |
12 | 11 | uneqri 4116 | 1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∖ cdif 3912 ∪ cun 3913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-dif 3918 df-un 3920 |
This theorem is referenced by: dfun3 4230 dfin3 4231 |
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