Theorem vneulem7 1137

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
vneulem7	((c ∩ a) ∪ (b ∪ d)) = (b ∪ d)

Proof of Theorem vneulem7

Step	Hyp	Ref	Expression
1		leao2 163	. . . . . 6 (c ∩ a) ≤ (a ∪ b)
2		leao1 162	. . . . . 6 (c ∩ a) ≤ (c ∪ d)
3	1, 2	ler2an 173	. . . . 5 (c ∩ a) ≤ ((a ∪ b) ∩ (c ∪ d))
4		vneulem6.1	. . . . 5 ((a ∪ b) ∩ (c ∪ d)) = 0
5	3, 4	lbtr 139	. . . 4 (c ∩ a) ≤ 0
6		le0 147	. . . 4 0 ≤ (c ∩ a)
7	5, 6	lebi 145	. . 3 (c ∩ a) = 0
8	7	ror 71	. 2 ((c ∩ a) ∪ (b ∪ d)) = (0 ∪ (b ∪ d))
9		or0r 103	. 2 (0 ∪ (b ∪ d)) = (b ∪ d)
10	8, 9	tr 62	1 ((c ∩ a) ∪ (b ∪ 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

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:  vneulem8  1138