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Theorem vneulem14 1142
 Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Hypothesis
Ref Expression
vneulem13.1 ((ab) ∩ (cd)) = 0
Assertion
Ref Expression
vneulem14 (((cd) ∪ (ab)) ∩ ((cd) ∪ (ab))) = ((cd) ∪ (ab))

Proof of Theorem vneulem14
StepHypRef Expression
1 vneulem12 1140 . 2 (((cd) ∪ (ab)) ∩ ((cd) ∪ (ab))) = ((cd) ∪ ((ab) ∩ ((cd) ∪ (ab))))
2 vneulem13.1 . . 3 ((ab) ∩ (cd)) = 0
32vneulem13 1141 . 2 ((cd) ∪ ((ab) ∩ ((cd) ∪ (ab)))) = ((cd) ∪ (ab))
41, 3tr 62 1 (((cd) ∪ (ab)) ∩ ((cd) ∪ (ab))) = ((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 1120 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  1144
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