Theorem dfun2 4245

Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 4246 and dfss4 4244 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.)

Assertion

Ref	Expression
dfun2	⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Proof of Theorem dfun2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3463	. . . . . . 7 ⊢ 𝑥 ∈ V
2		eldif 3936	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpbiran 709	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
4	3	anbi1i 624	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		eldif 3936	. . . . 5 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵))
6		ioran 985	. . . . 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 357	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
9		eldif 3936	. . . 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 278	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
12	11	uneqri 4131	1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ∖ cdif 3923   ∪ cun 3924

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931

This theorem is referenced by:  dfun3  4251  dfin3  4252