Theorem sklem 222

Description: Soundness lemma.

Hypothesis

Ref	Expression
sklem.1	a ≤ b

Assertion

Ref	Expression
sklem	(a⊥ ∪ b) = 1

Proof of Theorem sklem

Step	Hyp	Ref	Expression
1		or12 73	. . 3 (a⊥ ∪ (a ∪ b)) = (a ∪ (a⊥ ∪ b))
2		df-t 40	. . . . . 6 1 = (a ∪ a⊥ )
3	2	ax-r5 37	. . . . 5 (1 ∪ b) = ((a ∪ a⊥ ) ∪ b)
4	3	ax-r1 34	. . . 4 ((a ∪ a⊥ ) ∪ b) = (1 ∪ b)
5		ax-a3 31	. . . 4 ((a ∪ a⊥ ) ∪ b) = (a ∪ (a⊥ ∪ b))
6		ax-a2 30	. . . 4 (1 ∪ b) = (b ∪ 1)
7	4, 5, 6	3tr2 61	. . 3 (a ∪ (a⊥ ∪ b)) = (b ∪ 1)
8	1, 7	ax-r2 35	. 2 (a⊥ ∪ (a ∪ b)) = (b ∪ 1)
9		sklem.1	. . . 4 a ≤ b
10	9	df-le2 123	. . 3 (a ∪ b) = b
11	10	lor 66	. 2 (a⊥ ∪ (a ∪ b)) = (a⊥ ∪ b)
12		or1 96	. 2 (b ∪ 1) = 1
13	8, 11, 12	3tr2 61	1 (a⊥ ∪ b) = 1

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6  1wt 9

This theorem is referenced by:  ska13 233  ska15 236  lei3 238  oaidlem1 286  u1lemle1 692  u2lemle1 693  u3lemle1 694  u4lemle1 695  u5lemle1 696

This theorem was proved from axioms:  ax-a2 30  ax-a3 31  ax-a4 32  ax-r1 34  ax-r2 35  ax-r5 37

This theorem depends on definitions:  df-t 40  df-le2 123

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