Theorem l42modlem2 1150

Description: Lemma for l42mod 1151. (Contributed by NM, 8-Apr-2012.)

Assertion

Ref	Expression
l42modlem2	((((a ∪ b) ∩ c) ∪ d) ∩ e) ≤ (((a ∪ b) ∪ d) ∩ ((a ∪ b) ∪ e))

Proof of Theorem l42modlem2

Step	Hyp	Ref	Expression
1		lea 160	. . 3 ((a ∪ b) ∩ c) ≤ (a ∪ b)
2	1	leror 152	. 2 (((a ∪ b) ∩ c) ∪ d) ≤ ((a ∪ b) ∪ d)
3		leor 159	. 2 e ≤ ((a ∪ b) ∪ e)
4	2, 3	le2an 169	1 ((((a ∪ b) ∩ c) ∪ d) ∩ e) ≤ (((a ∪ b) ∪ d) ∩ ((a ∪ b) ∪ e))

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:  l42mod  1151