Theorem i5lei1 347

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

Assertion

Ref	Expression
i5lei1	(a →5 b) ≤ (a →1 b)

Proof of Theorem i5lei1

Step	Hyp	Ref	Expression
1		ax-a3 32	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
2		ax-a2 31	. . . 4 ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b))
3	1, 2	ax-r2 36	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b))
4		lea 160	. . . . 5 (a⊥ ∩ b) ≤ a⊥
5		lea 160	. . . . 5 (a⊥ ∩ b⊥ ) ≤ a⊥
6	4, 5	lel2or 170	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
7	6	leror 152	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∩ b))
8	3, 7	bltr 138	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ (a ∩ b))
9		df-i5 48	. 2 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
10		df-i1 44	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
11	8, 9, 10	le3tr1 140	1 (a →5 b) ≤ (a →1 b)

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₅ wi5 16

This proof depends on 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 proof 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 used by:  oago3.21x  890