Theorem i5lei2 348

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

Assertion

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

Proof of Theorem i5lei2

Step	Hyp	Ref	Expression
1		lear 161	. . . 4 (a ∩ b) ≤ b
2		lear 161	. . . 4 (a⊥ ∩ b) ≤ b
3	1, 2	lel2or 170	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ b
4	3	leror 152	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ≤ (b ∪ (a⊥ ∩ b⊥ ))
5		df-i5 48	. 2 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
6		df-i2 45	. 2 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
7	4, 5, 6	le3tr1 140	1 (a →5 b) ≤ (a →2 b)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13   →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-i2 45  df-i5 48  df-le1 130  df-le2 131

This theorem is referenced by:  oago3.21x  890  wdwom  1106