Theorem vneulem8 1138

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

Hypothesis

Ref	Expression
vneulem6.1	((a ∪ b) ∩ (c ∪ d)) = 0

Assertion

Ref	Expression
vneulem8	(((a ∪ b) ∪ d) ∩ ((b ∪ c) ∪ d)) = (b ∪ d)

Proof of Theorem vneulem8

Step	Hyp	Ref	Expression
1		vneulem6.1	. . 3 ((a ∪ b) ∩ (c ∪ d)) = 0
2	1	vneulem6 1136	. 2 (((a ∪ b) ∪ d) ∩ ((b ∪ c) ∪ d)) = ((c ∩ a) ∪ (b ∪ d))
3	1	vneulem7 1137	. 2 ((c ∩ a) ∪ (b ∪ d)) = (b ∪ d)
4	2, 3	tr 62	1 (((a ∪ b) ∪ d) ∩ ((b ∪ c) ∪ d)) = (b ∪ d)

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:  vneulem10  1140  vneulem15  1145