Theorem oa4lem2 938

Description: Lemma for 3-var to 4-var OA. (Contributed by NM, 27-Nov-1998.)

Hypotheses

Ref	Expression
oa4lem1.1	a ≤ b⊥
oa4lem1.2	c ≤ d⊥

Assertion

Ref	Expression
oa4lem2	(c ∪ d) ≤ ((a ∪ c)⊥ →2 d)

Proof of Theorem oa4lem2

Step	Hyp	Ref	Expression
1		leor 159	. . . . 5 c ≤ (a ∪ c)
2		ax-a1 30	. . . . 5 (a ∪ c) = (a ∪ c)⊥ ⊥
3	1, 2	lbtr 139	. . . 4 c ≤ (a ∪ c)⊥ ⊥
4		oa4lem1.2	. . . 4 c ≤ d⊥
5	3, 4	ler2an 173	. . 3 c ≤ ((a ∪ c)⊥ ⊥ ∩ d⊥ )
6	5	lelor 166	. 2 (d ∪ c) ≤ (d ∪ ((a ∪ c)⊥ ⊥ ∩ d⊥ ))
7		ax-a2 31	. 2 (c ∪ d) = (d ∪ c)
8		df-i2 45	. 2 ((a ∪ c)⊥ →2 d) = (d ∪ ((a ∪ c)⊥ ⊥ ∩ d⊥ ))
9	6, 7, 8	le3tr1 140	1 (c ∪ d) ≤ ((a ∪ c)⊥ →2 d)

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This proof depends on 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 proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131

This theorem is used by:  oa4lem3  939