Theorem sklem 230

Description: Soundness lemma. (Contributed by NM, 30-Aug-1997.)

Hypothesis

Ref	Expression
sklem.1	a ≤ b

Assertion

Ref	Expression
sklem	(a⊥ ∪ b) = 1

Proof of Theorem sklem

Step	Hyp	Ref	Expression
1		or12 80	. . 3 (a⊥ ∪ (a ∪ b)) = (a ∪ (a⊥ ∪ b))
2		df-t 41	. . . . . 6 1 = (a ∪ a⊥ )
3	2	ax-r5 38	. . . . 5 (1 ∪ b) = ((a ∪ a⊥ ) ∪ b)
4	3	ax-r1 35	. . . 4 ((a ∪ a⊥ ) ∪ b) = (1 ∪ b)
5		ax-a3 32	. . . 4 ((a ∪ a⊥ ) ∪ b) = (a ∪ (a⊥ ∪ b))
6		ax-a2 31	. . . 4 (1 ∪ b) = (b ∪ 1)
7	4, 5, 6	3tr2 64	. . 3 (a ∪ (a⊥ ∪ b)) = (b ∪ 1)
8	1, 7	ax-r2 36	. 2 (a⊥ ∪ (a ∪ b)) = (b ∪ 1)
9		sklem.1	. . . 4 a ≤ b
10	9	df-le2 131	. . 3 (a ∪ b) = b
11	10	lor 70	. 2 (a⊥ ∪ (a ∪ b)) = (a⊥ ∪ b)
12		or1 104	. 2 (b ∪ 1) = 1
13	8, 11, 12	3tr2 64	1 (a⊥ ∪ b) = 1

This proof depends on syntax axioms:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6  1wt 8

This proof depends on axioms:  ax-a2 31  ax-a3 32  ax-a4 33  ax-r1 35  ax-r2 36  ax-r5 38

This proof depends on definitions:  df-t 41  df-le2 131

This theorem is used by:  ska13  241  ska15  244  lei3  246  oaidlem1  294  id5id0  352  u1lemle1  710  u2lemle1  711  u3lemle1  712  u4lemle1  713  u5lemle1  714  lem3.3.3  1052  lem3.3.7i4e1  1069  lem3.3.7i4e2  1070  lem4.6.7  1103