Theorem u2lemonb 636

Description: Lemma for Dishkant implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

Ref	Expression
u2lemonb	((a →2 b) ∪ b⊥ ) = 1

Proof of Theorem u2lemonb

Step	Hyp	Ref	Expression
1		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r5 38	. 2 ((a →2 b) ∪ b⊥ ) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ )
3		or32 82	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = ((b ∪ b⊥ ) ∪ (a⊥ ∩ b⊥ ))
4		ax-a2 31	. . . 4 ((b ∪ b⊥ ) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ ))
5		df-t 41	. . . . . . 7 1 = (b ∪ b⊥ )
6	5	lor 70	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ 1) = ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ ))
7	6	ax-r1 35	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ 1)
8		or1 104	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ 1) = 1
9	7, 8	ax-r2 36	. . . 4 ((a⊥ ∩ b⊥ ) ∪ (b ∪ b⊥ )) = 1
10	4, 9	ax-r2 36	. . 3 ((b ∪ b⊥ ) ∪ (a⊥ ∩ b⊥ )) = 1
11	3, 10	ax-r2 36	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∪ b⊥ ) = 1
12	2, 11	ax-r2 36	1 ((a →2 b) ∪ b⊥ ) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₂ wi2 13

This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-a4 33  ax-r1 35  ax-r2 36  ax-r5 38

This theorem depends on definitions:  df-t 41  df-i2 45

This theorem is referenced by:  u2lemnab  651  u2lem3  750  oa23  936