Theorem wql1lem 279

Description: Classical implication inferred from Sakaki implication.

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 138	. 2 (a⊥ ∪ b) ≤ 1
2		wql1lem.1	. . . 4 (a →1 b) = 1
3	2	ax-r1 34	. . 3 1 = (a →1 b)
4		df-i1 43	. . . 4 (a →1 b) = (a⊥ ∪ (a ∩ b))
5		lear 153	. . . . 5 (a ∩ b) ≤ b
6	5	lelor 158	. . . 4 (a⊥ ∪ (a ∩ b)) ≤ (a⊥ ∪ b)
7	4, 6	bltr 130	. . 3 (a →1 b) ≤ (a⊥ ∪ b)
8	3, 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   →₁ wi1 13

This theorem is referenced by:  wql1 285

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-i1 43  df-le1 122  df-le2 123

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