Theorem i5lei4 342

Description: Relevance implication is l.e. non-tollens implication.

Assertion

Ref	Expression
i5lei4	(a →5 b) ≤ (a →4 b)

Proof of Theorem i5lei4

Step	Hyp	Ref	Expression
1		leo 150	. . . 4 a⊥ ≤ (a⊥ ∪ b)
2	1	leran 145	. . 3 (a⊥ ∩ b⊥ ) ≤ ((a⊥ ∪ b) ∩ b⊥ )
3	2	lelor 158	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ≤ (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
4		df-i5 47	. 2 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
5		df-i4 46	. 2 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
6	3, 4, 5	le3tr1 132	1 (a →5 b) ≤ (a →4 b)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 16   →₅ wi5 17

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i4 46  df-i5 47  df-le1 122  df-le2 123

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