Theorem dfun2 2233

Description: An alternate definition of the union of two classes in terms of class difference, requiring no dummy variables. Along with dfin2 2234 and dfss4 2232 it shows we can express union, intersection, and subset directly in terms of the single "primitive" operation \ (class difference).

Assertion

Ref	Expression
dfun2	|- (A u. B) = (V \ ((V \ A) \ B))

Proof of Theorem dfun2

Step	Hyp	Ref	Expression
1		eldif 2047	. . . . . . 7 |- (x e. (V \ A) <-> (x e. V /\ -. x e. A))
2		visset 1804	. . . . . . 7 |- x e. V
3	1, 2	mpbiran 726	. . . . . 6 |- (x e. (V \ A) <-> -. x e. A)
4	3	anbi1i 480	. . . . 5 |- ((x e. (V \ A) /\ -. x e. B) <-> (-. x e. A /\ -. x e. B))
5		eldif 2047	. . . . 5 |- (x e. ((V \ A) \ B) <-> (x e. (V \ A) /\ -. x e. B))
6		ioran 306	. . . . 5 |- (-. (x e. A \/ x e. B) <-> (-. x e. A /\ -. x e. B))
7	4, 5, 6	3bitr4 183	. . . 4 |- (x e. ((V \ A) \ B) <-> -. (x e. A \/ x e. B))
8	7	con2bii 221	. . 3 |- ((x e. A \/ x e. B) <-> -. x e. ((V \ A) \ B))
9		elun 2163	. . 3 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
10		eldif 2047	. . . 4 |- (x e. (V \ ((V \ A) \ B)) <-> (x e. V /\ -. x e. ((V \ A) \ B)))
11	10, 2	mpbiran 726	. . 3 |- (x e. (V \ ((V \ A) \ B)) <-> -. x e. ((V \ A) \ B))
12	8, 9, 11	3bitr4 183	. 2 |- (x e. (A u. B) <-> x e. (V \ ((V \ A) \ B)))
13	12	eqriv 1467	1 |- (A u. B) = (V \ ((V \ A) \ B))

Syntax hints:  -. wn 2   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802   \ cdif 2034   u. cun 2035

This theorem is referenced by:  dfun3 2236  dfin3 2237

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-dif 2039  df-un 2040

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