Theorem i5lei3 349

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

Assertion

Ref	Expression
i5lei3	(a →5 b) ≤ (a →3 b)

Proof of Theorem i5lei3

Step	Hyp	Ref	Expression
1		leor 159	. . . 4 b ≤ (a⊥ ∪ b)
2	1	lelan 167	. . 3 (a ∩ b) ≤ (a ∩ (a⊥ ∪ b))
3	2	leror 152	. 2 ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) ≤ ((a ∩ (a⊥ ∪ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
4		df-i5 48	. . 3 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
5		ax-a3 32	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
6	4, 5	ax-r2 36	. 2 (a →5 b) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
7		df-i3 46	. . 3 (a →3 b) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b)))
8		ax-a2 31	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ (a⊥ ∪ b))) = ((a ∩ (a⊥ ∪ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
9	7, 8	ax-r2 36	. 2 (a →3 b) = ((a ∩ (a⊥ ∪ b)) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
10	3, 6, 9	le3tr1 140	1 (a →5 b) ≤ (a →3 b)

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)