Theorem vneulem3 1133

Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96. (Contributed by NM, 15-Mar-2010.) (Revised by NM, 31-Mar-2011.)

Hypothesis

Ref	Expression
vneulem3.1	((x ∪ y) ∩ (u ∪ w)) = 0

Assertion

Ref	Expression
vneulem3	((((x ∪ y) ∩ (u ∪ w)) ∪ u) ∩ w) = (u ∩ w)

Proof of Theorem vneulem3

Step	Hyp	Ref	Expression
1		vneulem3.1	. . . 4 ((x ∪ y) ∩ (u ∪ w)) = 0
2	1	ror 71	. . 3 (((x ∪ y) ∩ (u ∪ w)) ∪ u) = (0 ∪ u)
3		or0r 103	. . 3 (0 ∪ u) = u
4	2, 3	tr 62	. 2 (((x ∪ y) ∩ (u ∪ w)) ∪ u) = u
5	4	ran 78	1 ((((x ∪ y) ∩ (u ∪ w)) ∪ u) ∩ w) = (u ∩ w)

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7  0wf 9

This theorem was proved from axioms:  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42

This theorem is referenced by:  vneulem4  1134