Theorem i5lei1 339

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

Assertion

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

Proof of Theorem i5lei1

Step	Hyp	Ref	Expression
1		ax-a3 31	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
2		ax-a2 30	. . . 4 ((a ∩ b) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b))
3	1, 2	ax-r2 35	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b))
4		lea 152	. . . . 5 (a⊥ ∩ b) ≤ a⊥
5		lea 152	. . . . 5 (a⊥ ∩ b⊥ ) ≤ a⊥
6	4, 5	lel2or 162	. . . 4 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ a⊥
7	6	leror 144	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ (a ∩ b)) ≤ (a⊥ ∪ (a ∩ b))
8	3, 7	bltr 130	. 2 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ )) ≤ (a⊥ ∪ (a ∩ b))
9		df-i5 47	. 2 (a →5 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ (a⊥ ∩ b⊥ ))
10		df-i1 43	. 2 (a →1 b) = (a⊥ ∪ (a ∩ b))
11	8, 9, 10	le3tr1 132	1 (a →5 b) ≤ (a →1 b)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →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-i1 43  df-i5 47  df-le1 122  df-le2 123

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