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Theorem i1com 708
 Description: Commutation expressed with →1 .
Hypothesis
Ref Expression
i1com.1 b ≤ (a1 b)
Assertion
Ref Expression
i1com a C b

Proof of Theorem i1com
StepHypRef Expression
1 ancom 74 . . . 4 (b ∩ (a1 b)) = ((a1 b) ∩ b)
2 i1com.1 . . . . 5 b ≤ (a1 b)
32df2le2 136 . . . 4 (b ∩ (a1 b)) = b
4 u1lemab 610 . . . . 5 ((a1 b) ∩ b) = ((ab) ∪ (ab))
5 ancom 74 . . . . . 6 (ab) = (ba)
6 ancom 74 . . . . . 6 (ab) = (ba )
75, 62or 72 . . . . 5 ((ab) ∪ (ab)) = ((ba) ∪ (ba ))
84, 7ax-r2 36 . . . 4 ((a1 b) ∩ b) = ((ba) ∪ (ba ))
91, 3, 83tr2 64 . . 3 b = ((ba) ∪ (ba ))
109df-c1 132 . 2 b C a
1110comcom 453 1 a C b
 Colors of variables: term Syntax hints:   ≤ wle 2   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  comanb  872
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