Theorem vneulem15 1145

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

Hypothesis

Ref	Expression
vneulem13.1	((a ∪ b) ∩ (c ∪ d)) = 0

Assertion

Ref	Expression
vneulem15	((a ∪ c) ∩ (b ∪ d)) = ((((a ∪ b) ∪ c) ∩ ((a ∪ c) ∪ d)) ∩ (((a ∪ b) ∪ d) ∩ ((b ∪ c) ∪ d)))

Proof of Theorem vneulem15

Step	Hyp	Ref	Expression
1		vneulem13.1	. . . 4 ((a ∪ b) ∩ (c ∪ d)) = 0
2	1	vneulem10 1140	. . 3 (((a ∪ b) ∪ c) ∩ ((a ∪ c) ∪ d)) = (a ∪ c)
3	1	vneulem8 1138	. . 3 (((a ∪ b) ∪ d) ∩ ((b ∪ c) ∪ d)) = (b ∪ d)
4	2, 3	2an 79	. 2 ((((a ∪ b) ∪ c) ∩ ((a ∪ c) ∪ d)) ∩ (((a ∪ b) ∪ d) ∩ ((b ∪ c) ∪ d))) = ((a ∪ c) ∩ (b ∪ d))
5	4	cm 61	1 ((a ∪ c) ∩ (b ∪ d)) = ((((a ∪ b) ∪ c) ∩ ((a ∪ c) ∪ d)) ∩ (((a ∪ b) ∪ d) ∩ ((b ∪ c) ∪ d)))

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:  vneulem  1147