Theorem leao 124

Description: Relation between two methods of expressing "less than or equal to". (Contributed by NM, 11-Aug-1997.)

Hypothesis

Ref	Expression
leao.1	(c ∩ b) = a

Assertion

Ref	Expression
leao	(a ∪ b) = b

Proof of Theorem leao

Step	Hyp	Ref	Expression
1		ax-a2 31	. . 3 (a ∪ b) = (b ∪ a)
2		leao.1	. . . . . 6 (c ∩ b) = a
3	2	ax-r1 35	. . . . 5 a = (c ∩ b)
4		ancom 74	. . . . . 6 (b ∩ c) = (c ∩ b)
5	4	ax-r1 35	. . . . 5 (c ∩ b) = (b ∩ c)
6	3, 5	ax-r2 36	. . . 4 a = (b ∩ c)
7	6	lor 70	. . 3 (b ∪ a) = (b ∪ (b ∩ c))
8	1, 7	ax-r2 36	. 2 (a ∪ b) = (b ∪ (b ∩ c))
9		orabs 120	. 2 (b ∪ (b ∩ c)) = b
10	8, 9	ax-r2 36	1 (a ∪ b) = b

This proof depends on syntax axioms:   = wb 1   ∪ wo 6   ∩ wa 7

This proof depends on axioms:  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40

This theorem is used by:  df2le1  135