Theorem vneulem4 1134

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
vneulem4	(((x ∪ y) ∪ u) ∩ w) = (u ∩ w)

Proof of Theorem vneulem4

Step	Hyp	Ref	Expression
1		vneulem1 1131	. 2 (((x ∪ y) ∪ u) ∩ w) = (((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w))
2		vneulem2 1132	. 2 (((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w)) = ((((x ∪ y) ∩ (u ∪ w)) ∪ u) ∩ w)
3		vneulem3.1	. . 3 ((x ∪ y) ∩ (u ∪ w)) = 0
4	3	vneulem3 1133	. 2 ((((x ∪ y) ∩ (u ∪ w)) ∪ u) ∩ w) = (u ∩ w)
5	1, 2, 4	3tr 65	1 (((x ∪ y) ∪ u) ∩ w) = (u ∩ w)

This proof depends on syntax axioms:   = wb 1   ∪ wo 6   ∩ wa 7  0wf 9

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1122

This proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is used by:  vneulem6  1136  vneulem9  1139