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Theorem vneulem11 1141
 Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 (Contributed by NM, 31-Mar-2011.)
Hypothesis
Ref Expression
vneulem6.1 ((ab) ∩ (cd)) = 0
Assertion
Ref Expression
vneulem11 (((bc) ∪ d) ∩ ((ac) ∪ d)) = ((cd) ∪ (ab))

Proof of Theorem vneulem11
StepHypRef Expression
1 ax-a3 32 . . . 4 ((bc) ∪ d) = (b ∪ (cd))
2 orcom 73 . . . 4 (b ∪ (cd)) = ((cd) ∪ b)
31, 2tr 62 . . 3 ((bc) ∪ d) = ((cd) ∪ b)
4 ax-a2 31 . . . . 5 (ac) = (ca)
54ror 71 . . . 4 ((ac) ∪ d) = ((ca) ∪ d)
6 or32 82 . . . 4 ((ca) ∪ d) = ((cd) ∪ a)
75, 6tr 62 . . 3 ((ac) ∪ d) = ((cd) ∪ a)
83, 72an 79 . 2 (((bc) ∪ d) ∩ ((ac) ∪ d)) = (((cd) ∪ b) ∩ ((cd) ∪ a))
9 ancom 74 . . . 4 ((cd) ∩ (ab)) = ((ab) ∩ (cd))
10 vneulem6.1 . . . 4 ((ab) ∩ (cd)) = 0
119, 10tr 62 . . 3 ((cd) ∩ (ab)) = 0
1211vneulem9 1139 . 2 (((cd) ∪ b) ∩ ((cd) ∪ a)) = ((ab) ∪ (cd))
13 orcom 73 . 2 ((ab) ∪ (cd)) = ((cd) ∪ (ab))
148, 12, 133tr 65 1 (((bc) ∪ d) ∩ ((ac) ∪ d)) = ((cd) ∪ (ab))
 Colors of variables: term 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:  vneulem16  1146
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