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Theorem vneulem4 1134
 Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96 (Contributed by NM, 15-Mar-2010.) (Revised by NM, 31-Mar-2011.)
Hypothesis
Ref Expression
vneulem3.1 ((xy) ∩ (uw)) = 0
Assertion
Ref Expression
vneulem4 (((xy) ∪ u) ∩ w) = (uw)

Proof of Theorem vneulem4
StepHypRef Expression
1 vneulem1 1131 . 2 (((xy) ∪ u) ∩ w) = (((xy) ∪ u) ∩ ((uw) ∩ w))
2 vneulem2 1132 . 2 (((xy) ∪ u) ∩ ((uw) ∩ w)) = ((((xy) ∩ (uw)) ∪ u) ∩ w)
3 vneulem3.1 . . 3 ((xy) ∩ (uw)) = 0
43vneulem3 1133 . 2 ((((xy) ∩ (uw)) ∪ u) ∩ w) = (uw)
51, 2, 43tr 65 1 (((xy) ∪ u) ∩ w) = (uw)
 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:  vneulem6  1136  vneulem9  1139
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