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Theorem comm0 178
 Description: Commutation with 0. Kalmbach 83 p. 20.
Assertion
Ref Expression
comm0 a C 0

Proof of Theorem comm0
StepHypRef Expression
1 ax-a2 31 . . . . 5 (0 ∪ a) = (a ∪ 0)
2 or0 102 . . . . 5 (a ∪ 0) = a
31, 2ax-r2 36 . . . 4 (0 ∪ a) = a
43ax-r1 35 . . 3 a = (0 ∪ a)
5 an0 108 . . . . 5 (a ∩ 0) = 0
6 df-f 42 . . . . . . . 8 0 = 1
76con2 67 . . . . . . 7 0 = 1
87lan 77 . . . . . 6 (a ∩ 0 ) = (a ∩ 1)
9 an1 106 . . . . . 6 (a ∩ 1) = a
108, 9ax-r2 36 . . . . 5 (a ∩ 0 ) = a
115, 102or 72 . . . 4 ((a ∩ 0) ∪ (a ∩ 0 )) = (0 ∪ a)
1211ax-r1 35 . . 3 (0 ∪ a) = ((a ∩ 0) ∪ (a ∩ 0 ))
134, 12ax-r2 36 . 2 a = ((a ∩ 0) ∪ (a ∩ 0 ))
1413df-c1 132 1 a C 0
 Colors of variables: term Syntax hints:   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  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-c1 132 This theorem is referenced by:  wcom0  407
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