Theorem wql1lem 287

Description: Classical implication inferred from Sakaki implication. (Contributed by NM, 5-Dec-1998.)

Hypothesis

Ref	Expression
wql1lem.1	(a →1 b) = 1

Assertion

Ref	Expression
wql1lem	(a⊥ ∪ b) = 1

Proof of Theorem wql1lem

Step	Hyp	Ref	Expression
1		le1 146	. 2 (a⊥ ∪ b) ≤ 1
2		wql1lem.1	. . . 4 (a →1 b) = 1
3	2	ax-r1 35	. . 3 1 = (a →1 b)
4		df-i1 44	. . . 4 (a →1 b) = (a⊥ ∪ (a ∩ b))
5		lear 161	. . . . 5 (a ∩ b) ≤ b
6	5	lelor 166	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ b)
7	4, 6	bltr 138	. . 3 (a →1 b) ≤ (a⊥ ∪ b)
8	3, 7	bltr 138	. 2 1 ≤ (a⊥ ∪ b)
9	1, 8	lebi 145	1 (a⊥ ∪ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  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-i1 44  df-le1 130  df-le2 131

This theorem is referenced by:  wql1  293  wdwom  1106