Theorem wler 391

Description: Add disjunct to right of l.e. (Contributed by NM, 13-Oct-1997.)

Hypothesis

Ref	Expression
wle.1	(a ≤2 b) = 1

Assertion

Ref	Expression
wler	(a ≤2 (b ∪ c)) = 1

Proof of Theorem wler

Step	Hyp	Ref	Expression
1		df-le 129	. 2 (a ≤2 (b ∪ c)) = ((a ∪ (b ∪ c)) ≡ (b ∪ c))
2		ax-a3 32	. . . . 5 ((a ∪ b) ∪ c) = (a ∪ (b ∪ c))
3	2	ax-r1 35	. . . 4 (a ∪ (b ∪ c)) = ((a ∪ b) ∪ c)
4	3	rbi 98	. . 3 ((a ∪ (b ∪ c)) ≡ (b ∪ c)) = (((a ∪ b) ∪ c) ≡ (b ∪ c))
5		df-le 129	. . . . . 6 (a ≤2 b) = ((a ∪ b) ≡ b)
6	5	ax-r1 35	. . . . 5 ((a ∪ b) ≡ b) = (a ≤2 b)
7		wle.1	. . . . 5 (a ≤2 b) = 1
8	6, 7	ax-r2 36	. . . 4 ((a ∪ b) ≡ b) = 1
9	8	wr5-2v 366	. . 3 (((a ∪ b) ∪ c) ≡ (b ∪ c)) = 1
10	4, 9	ax-r2 36	. 2 ((a ∪ (b ∪ c)) ≡ (b ∪ c)) = 1
11	1, 10	ax-r2 36	1 (a ≤2 (b ∪ c)) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6  1wt 8   ≤₂ wle2 10

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131

This theorem is referenced by: (None)