Theorem 0i1 273

Description: Antecedent of 0 on Sasaki conditional. (Contributed by NM, 24-Dec-1998.)

Assertion

Ref	Expression
0i1	(0 →1 a) = 1

Proof of Theorem 0i1

Step	Hyp	Ref	Expression
1		df-i1 44	. 2 (0 →1 a) = (0⊥ ∪ (0 ∩ a))
2		ax-a2 31	. . 3 (0⊥ ∪ (0 ∩ a)) = ((0 ∩ a) ∪ 0⊥ )
3		df-f 42	. . . . 5 0 = 1⊥
4	3	con2 67	. . . 4 0⊥ = 1
5	4	lor 70	. . 3 ((0 ∩ a) ∪ 0⊥ ) = ((0 ∩ a) ∪ 1)
6	2, 5	ax-r2 36	. 2 (0⊥ ∪ (0 ∩ a)) = ((0 ∩ a) ∪ 1)
7		or1 104	. 2 ((0 ∩ a) ∪ 1) = 1
8	1, 6, 7	3tr 65	1 (0 →1 a) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →₁ wi1 12

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-t 41  df-f 42  df-i1 44

This theorem is referenced by:  oa3-2lema  978  oa3-2to2s  990