Theorem vneulem2 1132

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

Assertion

Ref	Expression
vneulem2	(((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w)) = ((((x ∪ y) ∩ (u ∪ w)) ∪ u) ∩ w)

Proof of Theorem vneulem2

Step	Hyp	Ref	Expression
1		anass 76	. . 3 ((((x ∪ y) ∪ u) ∩ (u ∪ w)) ∩ w) = (((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w))
2	1	cm 61	. 2 (((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w)) = ((((x ∪ y) ∪ u) ∩ (u ∪ w)) ∩ w)
3		ax-a2 31	. . . . 5 ((x ∪ y) ∪ u) = (u ∪ (x ∪ y))
4	3	ran 78	. . . 4 (((x ∪ y) ∪ u) ∩ (u ∪ w)) = ((u ∪ (x ∪ y)) ∩ (u ∪ w))
5		ml 1123	. . . . 5 (u ∪ ((x ∪ y) ∩ (u ∪ w))) = ((u ∪ (x ∪ y)) ∩ (u ∪ w))
6	5	cm 61	. . . 4 ((u ∪ (x ∪ y)) ∩ (u ∪ w)) = (u ∪ ((x ∪ y) ∩ (u ∪ w)))
7		orcom 73	. . . 4 (u ∪ ((x ∪ y) ∩ (u ∪ w))) = (((x ∪ y) ∩ (u ∪ w)) ∪ u)
8	4, 6, 7	3tr 65	. . 3 (((x ∪ y) ∪ u) ∩ (u ∪ w)) = (((x ∪ y) ∩ (u ∪ w)) ∪ u)
9	8	ran 78	. 2 ((((x ∪ y) ∪ u) ∩ (u ∪ w)) ∩ w) = ((((x ∪ y) ∩ (u ∪ w)) ∪ u) ∩ w)
10	2, 9	tr 62	1 (((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w)) = ((((x ∪ y) ∩ (u ∪ w)) ∪ u) ∩ w)

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

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:  vneulem4  1134