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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 ((ab) ∩ (cd)) = 0
Assertion
Ref Expression
vneulem10 (((ab) ∪ c) ∩ ((ac) ∪ d)) = (ac)

Proof of Theorem vneulem10
StepHypRef Expression
1 ax-a2 31 . . . 4 (ab) = (ba)
21ax-r5 38 . . 3 ((ab) ∪ c) = ((ba) ∪ c)
3 or32 82 . . 3 ((ac) ∪ d) = ((ad) ∪ c)
42, 32an 79 . 2 (((ab) ∪ c) ∩ ((ac) ∪ d)) = (((ba) ∪ c) ∩ ((ad) ∪ c))
5 orcom 73 . . . . 5 (ba) = (ab)
6 orcom 73 . . . . 5 (dc) = (cd)
75, 62an 79 . . . 4 ((ba) ∩ (dc)) = ((ab) ∩ (cd))
8 vneulem6.1 . . . 4 ((ab) ∩ (cd)) = 0
97, 8tr 62 . . 3 ((ba) ∩ (dc)) = 0
109vneulem8 1138 . 2 (((ba) ∪ c) ∩ ((ad) ∪ c)) = (ac)
114, 10tr 62 1 (((ab) ∪ c) ∩ ((ac) ∪ d)) = (ac)
 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:  vneulem15  1145
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