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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 ((ab) ∩ (cd)) = 0
Assertion
Ref Expression
vneulem7 ((ca) ∪ (bd)) = (bd)

Proof of Theorem vneulem7
StepHypRef Expression
1 leao2 163 . . . . . 6 (ca) ≤ (ab)
2 leao1 162 . . . . . 6 (ca) ≤ (cd)
31, 2ler2an 173 . . . . 5 (ca) ≤ ((ab) ∩ (cd))
4 vneulem6.1 . . . . 5 ((ab) ∩ (cd)) = 0
53, 4lbtr 139 . . . 4 (ca) ≤ 0
6 le0 147 . . . 4 0 ≤ (ca)
75, 6lebi 145 . . 3 (ca) = 0
87ror 71 . 2 ((ca) ∪ (bd)) = (0 ∪ (bd))
9 or0r 103 . 2 (0 ∪ (bd)) = (bd)
108, 9tr 62 1 ((ca) ∪ (bd)) = (bd)
 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 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:  vneulem8  1138
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