Theorem lel2an 171

Description: Conjunction of 2 l.e.'s. (Contributed by NM, 11-Nov-1997.)

Hypotheses

Ref	Expression
lel2.1	a ≤ b
lel2.2	c ≤ b

Assertion

Ref	Expression
lel2an	(a ∩ c) ≤ b

Proof of Theorem lel2an

Step	Hyp	Ref	Expression
1		lel2.1	. . 3 a ≤ b
2		lel2.2	. . 3 c ≤ b
3	1, 2	le2an 169	. 2 (a ∩ c) ≤ (b ∩ b)
4		anidm 111	. 2 (b ∩ b) = b
5	3, 4	lbtr 139	1 (a ∩ c) ≤ b

Syntax hints:   ≤ wle 2   ∩ wa 7

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by: (None)