Theorem wql2lem3 290

Description: Lemma for →₂ WQL axiom. (Contributed by NM, 6-Dec-1998.)

Hypothesis

Ref	Expression
wql2lem3.1	(a →2 b) = 1

Assertion

Ref	Expression
wql2lem3	((a ∩ b⊥ ) →2 a⊥ ) = 1

Proof of Theorem wql2lem3

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 ((a ∩ b⊥ ) →2 a⊥ ) = (a⊥ ∪ ((a ∩ b⊥ )⊥ ∩ a⊥ ⊥ ))
2		oran2 92	. . . . . 6 (a⊥ ∪ b) = (a ∩ b⊥ )⊥
3	2	ax-r1 35	. . . . 5 (a ∩ b⊥ )⊥ = (a⊥ ∪ b)
4	3	ran 78	. . . 4 ((a ∩ b⊥ )⊥ ∩ a⊥ ⊥ ) = ((a⊥ ∪ b) ∩ a⊥ ⊥ )
5		ancom 74	. . . 4 ((a⊥ ∪ b) ∩ a⊥ ⊥ ) = (a⊥ ⊥ ∩ (a⊥ ∪ b))
6	4, 5	ax-r2 36	. . 3 ((a ∩ b⊥ )⊥ ∩ a⊥ ⊥ ) = (a⊥ ⊥ ∩ (a⊥ ∪ b))
7	6	lor 70	. 2 (a⊥ ∪ ((a ∩ b⊥ )⊥ ∩ a⊥ ⊥ )) = (a⊥ ∪ (a⊥ ⊥ ∩ (a⊥ ∪ b)))
8		wql2lem3.1	. . . 4 (a →2 b) = 1
9	8	wql2lem 288	. . 3 (a⊥ ∪ b) = 1
10		omlem2 128	. . 3 ((a⊥ ∪ b)⊥ ∪ (a⊥ ∪ (a⊥ ⊥ ∩ (a⊥ ∪ b)))) = 1
11	9, 10	skr0 242	. 2 (a⊥ ∪ (a⊥ ⊥ ∩ (a⊥ ∪ b))) = 1
12	1, 7, 11	3tr 65	1 ((a ∩ b⊥ ) →2 a⊥ ) = 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: (None)