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Theorem i5lei1 347
 Description: Relevance implication is l.e. Sasaki implication.
Assertion
Ref Expression
i5lei1 (a5 b) ≤ (a1 b)

Proof of Theorem i5lei1
StepHypRef Expression
1 ax-a3 32 . . . 4 (((ab) ∪ (ab)) ∪ (ab )) = ((ab) ∪ ((ab) ∪ (ab )))
2 ax-a2 31 . . . 4 ((ab) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ (ab))
31, 2ax-r2 36 . . 3 (((ab) ∪ (ab)) ∪ (ab )) = (((ab) ∪ (ab )) ∪ (ab))
4 lea 160 . . . . 5 (ab) ≤ a
5 lea 160 . . . . 5 (ab ) ≤ a
64, 5lel2or 170 . . . 4 ((ab) ∪ (ab )) ≤ a
76leror 152 . . 3 (((ab) ∪ (ab )) ∪ (ab)) ≤ (a ∪ (ab))
83, 7bltr 138 . 2 (((ab) ∪ (ab)) ∪ (ab )) ≤ (a ∪ (ab))
9 df-i5 48 . 2 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
10 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
118, 9, 10le3tr1 140 1 (a5 b) ≤ (a1 b)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →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-i1 44  df-i5 48  df-le1 130  df-le2 131 This theorem is referenced by:  oago3.21x  890
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