Theorem str 189

Description: Strengthening rule. (Contributed by NM, 18-Nov-1998.)

Hypotheses

Ref	Expression
str.1	a ≤ (b ∪ c)
str.2	(a ∩ (b ∪ c)) ≤ b

Assertion

Ref	Expression
str	a ≤ b

Proof of Theorem str

Step	Hyp	Ref	Expression
1		id 59	. . . 4 a = a
2	1	bile 142	. . 3 a ≤ a
3		str.1	. . 3 a ≤ (b ∪ c)
4	2, 3	ler2an 173	. 2 a ≤ (a ∩ (b ∪ c))
5		str.2	. 2 (a ∩ (b ∪ c)) ≤ b
6	4, 5	letr 137	1 a ≤ b

Syntax hints:   ≤ wle 2   ∪ wo 6   ∩ 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)