Theorem wql2lem 288

Description: Classical implication inferred from Dishkant implication. (Contributed by NM, 6-Dec-1998.)

Hypothesis

Ref	Expression
wql2lem.1	(a →2 b) = 1

Assertion

Ref	Expression
wql2lem	(a⊥ ∪ b) = 1

Proof of Theorem wql2lem

Step	Hyp	Ref	Expression
1		le1 146	. 2 (a⊥ ∪ b) ≤ 1
2		df-i2 45	. . . 4 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
3		wql2lem.1	. . . 4 (a →2 b) = 1
4		ax-a2 31	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
5	2, 3, 4	3tr2 64	. . 3 1 = ((a⊥ ∩ b⊥ ) ∪ b)
6		lea 160	. . . 4 (a⊥ ∩ b⊥ ) ≤ a⊥
7	6	leror 152	. . 3 ((a⊥ ∩ b⊥ ) ∪ b) ≤ (a⊥ ∪ b)
8	5, 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   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  wql2lem3  290