Theorem leoa 123

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

Hypothesis

Ref	Expression
leoa.1	(a ∪ c) = b

Assertion

Ref	Expression
leoa	(a ∩ b) = a

Proof of Theorem leoa

Step	Hyp	Ref	Expression
1		leoa.1	. . . 4 (a ∪ c) = b
2	1	ax-r1 35	. . 3 b = (a ∪ c)
3	2	lan 77	. 2 (a ∩ b) = (a ∩ (a ∪ c))
4		anabs 121	. 2 (a ∩ (a ∪ c)) = a
5	3, 4	ax-r2 36	1 (a ∩ b) = a

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7

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

This theorem depends on definitions:  df-a 40

This theorem is referenced by:  df2le2  136  wlem3.1  210