Theorem u2lemona 608

Description: Lemma for Dishkant implication study.

Assertion

Ref	Expression
u2lemona	((a →2 b) ∪ a⊥ ) = (a⊥ ∪ b)

Proof of Theorem u2lemona

Step	Hyp	Ref	Expression
1		df-i2 44	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2	1	ax-r5 37	. 2 ((a →2 b) ∪ a⊥ ) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ )
3		ax-a3 31	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = (b ∪ ((a⊥ ∩ b⊥ ) ∪ a⊥ ))
4		ax-a2 30	. . . 4 (b ∪ ((a⊥ ∩ b⊥ ) ∪ a⊥ )) = (((a⊥ ∩ b⊥ ) ∪ a⊥ ) ∪ b)
5		lea 152	. . . . . 6 (a⊥ ∩ b⊥ ) ≤ a⊥
6	5	df-le2 123	. . . . 5 ((a⊥ ∩ b⊥ ) ∪ a⊥ ) = a⊥
7	6	ax-r5 37	. . . 4 (((a⊥ ∩ b⊥ ) ∪ a⊥ ) ∪ b) = (a⊥ ∪ b)
8	4, 7	ax-r2 35	. . 3 (b ∪ ((a⊥ ∩ b⊥ ) ∪ a⊥ )) = (a⊥ ∪ b)
9	3, 8	ax-r2 35	. 2 ((b ∪ (a⊥ ∩ b⊥ )) ∪ a⊥ ) = (a⊥ ∪ b)
10	2, 9	ax-r2 35	1 ((a →2 b) ∪ a⊥ ) = (a⊥ ∪ b)

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

This theorem is referenced by:  u2lemnaa 623

This theorem was proved from axioms:  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r5 37

This theorem depends on definitions:  df-a 39  df-i2 44  df-le1 122  df-le2 123

metamath.org