Theorem vneulem10 1140

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
vneulem10	(((a ∪ b) ∪ c) ∩ ((a ∪ c) ∪ d)) = (a ∪ c)

Proof of Theorem vneulem10

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . 4 (a ∪ b) = (b ∪ a)
2	1	ax-r5 38	. . 3 ((a ∪ b) ∪ c) = ((b ∪ a) ∪ c)
3		or32 82	. . 3 ((a ∪ c) ∪ d) = ((a ∪ d) ∪ c)
4	2, 3	2an 79	. 2 (((a ∪ b) ∪ c) ∩ ((a ∪ c) ∪ d)) = (((b ∪ a) ∪ c) ∩ ((a ∪ d) ∪ c))
5		orcom 73	. . . . 5 (b ∪ a) = (a ∪ b)
6		orcom 73	. . . . 5 (d ∪ c) = (c ∪ d)
7	5, 6	2an 79	. . . 4 ((b ∪ a) ∩ (d ∪ c)) = ((a ∪ b) ∩ (c ∪ d))
8		vneulem6.1	. . . 4 ((a ∪ b) ∩ (c ∪ d)) = 0
9	7, 8	tr 62	. . 3 ((b ∪ a) ∩ (d ∪ c)) = 0
10	9	vneulem8 1138	. 2 (((b ∪ a) ∪ c) ∩ ((a ∪ d) ∪ c)) = (a ∪ c)
11	4, 10	tr 62	1 (((a ∪ b) ∪ c) ∩ ((a ∪ c) ∪ d)) = (a ∪ c)

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

This theorem was proved from 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 theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by:  vneulem15  1145