Theorem i5lei2 340

Description: Relevance implication is l.e. Dishkant implication.

Assertion

Ref	Expression
i5lei2	(a →5 b) ≤ (a →2 b)

Proof of Theorem i5lei2

Step	Hyp	Ref	Expression
1		lear 153	. . . 4 (a ∩ b) ≤ b
2		lear 153	. . . 4 (a⊥ ∩ b) ≤ b
3	1, 2	lel2or 162	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ b
4	3	leror 144	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ≤ (b ∪ (a⊥ ∩ b⊥ ))
5		df-i5 47	. 2 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
6		df-i2 44	. 2 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
7	4, 5, 6	le3tr1 132	1 (a →5 b) ≤ (a →2 b)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14   →₅ wi5 17

This theorem is referenced by:  oago3.21x 872

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-i2 44  df-i5 47  df-le1 122  df-le2 123

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