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Theorem i5lei3 349
 Description: Relevance implication is l.e. Kalmbach implication.
Assertion
Ref Expression
i5lei3 (a5 b) ≤ (a3 b)

Proof of Theorem i5lei3
StepHypRef Expression
1 leor 159 . . . 4 b ≤ (ab)
21lelan 167 . . 3 (ab) ≤ (a ∩ (ab))
32leror 152 . 2 ((ab) ∪ ((ab) ∪ (ab ))) ≤ ((a ∩ (ab)) ∪ ((ab) ∪ (ab )))
4 df-i5 48 . . 3 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
5 ax-a3 32 . . 3 (((ab) ∪ (ab)) ∪ (ab )) = ((ab) ∪ ((ab) ∪ (ab )))
64, 5ax-r2 36 . 2 (a5 b) = ((ab) ∪ ((ab) ∪ (ab )))
7 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
8 ax-a2 31 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) = ((a ∩ (ab)) ∪ ((ab) ∪ (ab )))
97, 8ax-r2 36 . 2 (a3 b) = ((a ∩ (ab)) ∪ ((ab) ∪ (ab )))
103, 6, 9le3tr1 140 1 (a5 b) ≤ (a3 b)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14   →5 wi5 16 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-i3 46  df-i5 48  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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