Theorem dfun3 2246

Description: Union defined in terms of intersection (DeMorgan's law). Definition of union in [Mendelson] p. 231.

Assertion

Ref	Expression
dfun3	|- (A u. B) = (V \ ((V \ A) i^i (V \ B)))

Proof of Theorem dfun3

Step	Hyp	Ref	Expression
1		dfun2 2243	. 2 |- (A u. B) = (V \ ((V \ A) \ B))
2		dfin2 2244	. . . 4 |- ((V \ A) i^i (V \ B)) = ((V \ A) \ (V \ (V \ B)))
3		ddif 2169	. . . . 5 |- (V \ (V \ B)) = B
4	3	difeq2i 2156	. . . 4 |- ((V \ A) \ (V \ (V \ B))) = ((V \ A) \ B)
5	2, 4	eqtr2 1496	. . 3 |- ((V \ A) \ B) = ((V \ A) i^i (V \ B))
6	5	difeq2i 2156	. 2 |- (V \ ((V \ A) \ B)) = (V \ ((V \ A) i^i (V \ B)))
7	1, 6	eqtr 1495	1 |- (A u. B) = (V \ ((V \ A) i^i (V \ B)))

Syntax hints:   = wceq 956  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046

This theorem is referenced by:  difundi 2257  undifv 2339

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-dif 2049  df-un 2050  df-in 2051

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