Theorem wletr 396

Description: Transitive law for l.e. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
wletr.1	(a ≤2 b) = 1
wletr.2	(b ≤2 c) = 1

Assertion

Ref	Expression
wletr	(a ≤2 c) = 1

Proof of Theorem wletr

Step	Hyp	Ref	Expression
1		wletr.1	. . . . . . . 8 (a ≤2 b) = 1
2	1	wdf-le2 379	. . . . . . 7 ((a ∪ b) ≡ b) = 1
3	2	wr5-2v 366	. . . . . 6 (((a ∪ b) ∪ c) ≡ (b ∪ c)) = 1
4	3	wr1 197	. . . . 5 ((b ∪ c) ≡ ((a ∪ b) ∪ c)) = 1
5		wletr.2	. . . . . 6 (b ≤2 c) = 1
6	5	wdf-le2 379	. . . . 5 ((b ∪ c) ≡ c) = 1
7		ax-a3 32	. . . . . 6 ((a ∪ b) ∪ c) = (a ∪ (b ∪ c))
8	7	bi1 118	. . . . 5 (((a ∪ b) ∪ c) ≡ (a ∪ (b ∪ c))) = 1
9	4, 6, 8	w3tr2 375	. . . 4 (c ≡ (a ∪ (b ∪ c))) = 1
10	9	wlan 370	. . 3 ((a ∩ c) ≡ (a ∩ (a ∪ (b ∪ c)))) = 1
11		anabs 121	. . . 4 (a ∩ (a ∪ (b ∪ c))) = a
12	11	bi1 118	. . 3 ((a ∩ (a ∪ (b ∪ c))) ≡ a) = 1
13	10, 12	wr2 371	. 2 ((a ∩ c) ≡ a) = 1
14	13	wdf2le1 385	1 (a ≤2 c) = 1

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7  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:  wle2or  403  wle2an  404  ska4  433