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Theorem vneulem16 1144
Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96
Hypothesis
Ref Expression
vneulem13.1 ((ab) ∩ (cd)) = 0
Assertion
Ref Expression
vneulem16 ((((ab) ∪ c) ∩ ((ac) ∪ d)) ∩ (((ab) ∪ d) ∩ ((bc) ∪ d))) = ((ab) ∪ (cd))

Proof of Theorem vneulem16
StepHypRef Expression
1 ancom 74 . 2 ((((ab) ∪ c) ∩ ((ac) ∪ d)) ∩ (((ab) ∪ d) ∩ ((bc) ∪ d))) = ((((ab) ∪ d) ∩ ((bc) ∪ d)) ∩ (((ab) ∪ c) ∩ ((ac) ∪ d)))
2 an4 86 . . 3 ((((ab) ∪ d) ∩ ((bc) ∪ d)) ∩ (((ab) ∪ c) ∩ ((ac) ∪ d))) = ((((ab) ∪ d) ∩ ((ab) ∪ c)) ∩ (((bc) ∪ d) ∩ ((ac) ∪ d)))
3 vneulem13.1 . . . . 5 ((ab) ∩ (cd)) = 0
43vneulem9 1137 . . . 4 (((ab) ∪ d) ∩ ((ab) ∪ c)) = ((cd) ∪ (ab))
53vneulem11 1139 . . . 4 (((bc) ∪ d) ∩ ((ac) ∪ d)) = ((cd) ∪ (ab))
64, 52an 79 . . 3 ((((ab) ∪ d) ∩ ((ab) ∪ c)) ∩ (((bc) ∪ d) ∩ ((ac) ∪ d))) = (((cd) ∪ (ab)) ∩ ((cd) ∪ (ab)))
72, 6tr 62 . 2 ((((ab) ∪ d) ∩ ((bc) ∪ d)) ∩ (((ab) ∪ c) ∩ ((ac) ∪ d))) = (((cd) ∪ (ab)) ∩ ((cd) ∪ (ab)))
83vneulem14 1142 . . 3 (((cd) ∪ (ab)) ∩ ((cd) ∪ (ab))) = ((cd) ∪ (ab))
9 orcom 73 . . 3 ((cd) ∪ (ab)) = ((ab) ∪ (cd))
108, 9tr 62 . 2 (((cd) ∪ (ab)) ∩ ((cd) ∪ (ab))) = ((ab) ∪ (cd))
111, 7, 103tr 65 1 ((((ab) ∪ c) ∩ ((ac) ∪ d)) ∩ (((ab) ∪ d) ∩ ((bc) ∪ d))) = ((ab) ∪ (cd))
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:  vneulem  1145
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