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Theorem comi1 709
 Description: Commutation expressed with →1 .
Hypothesis
Ref Expression
comi1.1 a C b
Assertion
Ref Expression
comi1 b ≤ (a1 b)

Proof of Theorem comi1
StepHypRef Expression
1 ancom 74 . . . . 5 (ba) = (ab)
21ax-r5 38 . . . 4 ((ba) ∪ (ba )) = ((ab) ∪ (ba ))
3 ax-a2 31 . . . 4 ((ab) ∪ (ba )) = ((ba ) ∪ (ab))
42, 3ax-r2 36 . . 3 ((ba) ∪ (ba )) = ((ba ) ∪ (ab))
5 lear 161 . . . 4 (ba ) ≤ a
65leror 152 . . 3 ((ba ) ∪ (ab)) ≤ (a ∪ (ab))
74, 6bltr 138 . 2 ((ba) ∪ (ba )) ≤ (a ∪ (ab))
8 comi1.1 . . . 4 a C b
98comcom 453 . . 3 b C a
109df-c2 133 . 2 b = ((ba) ∪ (ba ))
11 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
127, 10, 11le3tr1 140 1 b ≤ (a1 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-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: (None)
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