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Theorem i5lei3 349
Description: Relevance implication is less than or equal to Kalmbach implication. (Contributed by NM, 26-Jun-2003.)
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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