Theorem u1lemc6 706

Description: Commutation theorem for Sasaki implication. (Contributed by NM, 19-Mar-1999.)

Assertion

Ref	Expression
u1lemc6	(a →1 b) C (a⊥ →1 b)

Proof of Theorem u1lemc6

Step	Hyp	Ref	Expression
1		lea 160	. . . . . 6 (a ∩ (a⊥ ∪ b⊥ )) ≤ a
2		ax-a1 30	. . . . . 6 a = a⊥ ⊥
3	1, 2	lbtr 139	. . . . 5 (a ∩ (a⊥ ∪ b⊥ )) ≤ a⊥ ⊥
4		leo 158	. . . . 5 a⊥ ⊥ ≤ (a⊥ ⊥ ∪ (a⊥ ∩ b))
5	3, 4	letr 137	. . . 4 (a ∩ (a⊥ ∪ b⊥ )) ≤ (a⊥ ⊥ ∪ (a⊥ ∩ b))
6		ud1lem0c 277	. . . 4 (a →1 b)⊥ = (a ∩ (a⊥ ∪ b⊥ ))
7		df-i1 44	. . . 4 (a⊥ →1 b) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
8	5, 6, 7	le3tr1 140	. . 3 (a →1 b)⊥ ≤ (a⊥ →1 b)
9	8	lecom 180	. 2 (a →1 b)⊥ C (a⊥ →1 b)
10	9	comcom6 459	1 (a →1 b) C (a⊥ →1 b)

Syntax hints:   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  negantlem2  849