Theorem ler2or 172

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

Hypotheses

Ref	Expression
ler2.1	a ≤ b
ler2.2	a ≤ c

Assertion

Ref	Expression
ler2or	a ≤ (b ∪ c)

Proof of Theorem ler2or

Step	Hyp	Ref	Expression
1		oridm 110	. . 3 (a ∪ a) = a
2	1	ax-r1 35	. 2 a = (a ∪ a)
3		ler2.1	. . 3 a ≤ b
4		ler2.2	. . 3 a ≤ c
5	3, 4	le2or 168	. 2 (a ∪ a) ≤ (b ∪ c)
6	2, 5	bltr 138	1 a ≤ (b ∪ c)

Syntax hints:   ≤ wle 2   ∪ wo 6

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:  distid  887