Theorem wle2or 403

Description: Disjunction of 2 l.e.'s. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
wle2.1	(a ≤2 b) = 1
wle2.2	(c ≤2 d) = 1

Assertion

Ref	Expression
wle2or	((a ∪ c) ≤2 (b ∪ d)) = 1

Proof of Theorem wle2or

Step	Hyp	Ref	Expression
1		wle2.1	. . 3 (a ≤2 b) = 1
2	1	wleror 393	. 2 ((a ∪ c) ≤2 (b ∪ c)) = 1
3		wle2.2	. . . 4 (c ≤2 d) = 1
4	3	wleror 393	. . 3 ((c ∪ b) ≤2 (d ∪ b)) = 1
5		ax-a2 31	. . . 4 (b ∪ c) = (c ∪ b)
6	5	bi1 118	. . 3 ((b ∪ c) ≡ (c ∪ b)) = 1
7		ax-a2 31	. . . 4 (b ∪ d) = (d ∪ b)
8	7	bi1 118	. . 3 ((b ∪ d) ≡ (d ∪ b)) = 1
9	4, 6, 8	wle3tr1 399	. 2 ((b ∪ c) ≤2 (b ∪ d)) = 1
10	2, 9	wletr 396	1 ((a ∪ c) ≤2 (b ∪ d)) = 1

Syntax hints:   = wb 1   ∪ 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:  wledi  405  wledio  406