Theorem dfun2 4190

Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 4191 and dfss4 4189 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 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
2		eldif 3893	. . . . . . 7 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpbiran 705	. . . . . 6 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
4	3	anbi1i 623	. . . . 5 ⊢ ((𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
5		eldif 3893	. . . . 5 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐴) ∧ ¬ 𝑥 ∈ 𝐵))
6		ioran 980	. . . . 5 ⊢ (¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 𝐵))
7	4, 5, 6	3bitr4i 302	. . . 4 ⊢ (𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵) ↔ ¬ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
8	7	con2bii 357	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
9		eldif 3893	. . . 4 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵)))
10	1, 9	mpbiran 705	. . 3 ⊢ (𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)) ↔ ¬ 𝑥 ∈ ((V ∖ 𝐴) ∖ 𝐵))
11	8, 10	bitr4i 277	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (V ∖ ((V ∖ 𝐴) ∖ 𝐵)))
12	11	uneqri 4081	1 ⊢ (𝐴 ∪ 𝐵) = (V ∖ ((V ∖ 𝐴) ∖ 𝐵))

Syntax hints:  ¬ wn 3   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888

This theorem is referenced by:  dfun3  4196  dfin3  4197