Theorem wql2lem 280

Description: Classical implication inferred from Dishkant implication.

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 138	. 2 (a⊥ ∪ b) ≤ 1
2		df-i2 44	. . . 4 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
3		wql2lem.1	. . . 4 (a →2 b) = 1
4		ax-a2 30	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ b)
5	2, 3, 4	3tr2 61	. . 3 1 = ((a⊥ ∩ b⊥ ) ∪ b)
6		lea 152	. . . 4 (a⊥ ∩ b⊥ ) ≤ a⊥
7	6	leror 144	. . 3 ((a⊥ ∩ b⊥ ) ∪ b) ≤ (a⊥ ∪ b)
8	5, 7	bltr 130	. 2 1 ≤ (a⊥ ∪ b)
9	1, 8	lebi 137	1 (a⊥ ∪ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₂ wi2 14

This theorem is referenced by:  wql2lem3 282

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  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-i2 44  df-le1 122  df-le2 123

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