Theorem dfun2 3490

Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 3491 and dfss4 3489 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 2862	. . . . . . 7 ⊢ x ∈ V
2		eldif 3221	. . . . . . 7 ⊢ (x ∈ (V ∖ A) ↔ (x ∈ V ∧ ¬ x ∈ A))
3	1, 2	mpbiran 884	. . . . . 6 ⊢ (x ∈ (V ∖ A) ↔ ¬ x ∈ A)
4	3	anbi1i 676	. . . . 5 ⊢ ((x ∈ (V ∖ A) ∧ ¬ x ∈ B) ↔ (¬ x ∈ A ∧ ¬ x ∈ B))
5		eldif 3221	. . . . 5 ⊢ (x ∈ ((V ∖ A) ∖ B) ↔ (x ∈ (V ∖ A) ∧ ¬ x ∈ B))
6		ioran 476	. . . . 5 ⊢ (¬ (x ∈ A ∨ x ∈ B) ↔ (¬ x ∈ A ∧ ¬ x ∈ B))
7	4, 5, 6	3bitr4i 268	. . . 4 ⊢ (x ∈ ((V ∖ A) ∖ B) ↔ ¬ (x ∈ A ∨ x ∈ B))
8	7	con2bii 322	. . 3 ⊢ ((x ∈ A ∨ x ∈ B) ↔ ¬ x ∈ ((V ∖ A) ∖ B))
9		eldif 3221	. . . 4 ⊢ (x ∈ (V ∖ ((V ∖ A) ∖ B)) ↔ (x ∈ V ∧ ¬ x ∈ ((V ∖ A) ∖ B)))
10	1, 9	mpbiran 884	. . 3 ⊢ (x ∈ (V ∖ ((V ∖ A) ∖ B)) ↔ ¬ x ∈ ((V ∖ A) ∖ B))
11	8, 10	bitr4i 243	. 2 ⊢ ((x ∈ A ∨ x ∈ B) ↔ x ∈ (V ∖ ((V ∖ A) ∖ B)))
12	11	uneqri 3406	1 ⊢ (A ∪ B) = (V ∖ ((V ∖ A) ∖ B))

Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206   ∪ cun 3207

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215

This theorem is referenced by:  dfun3  3493  dfin3  3494