Theorem oa4lem1 917

Description: Lemma for 3-var to 4-var OA.

Hypotheses

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

Assertion

Ref	Expression
oa4lem1	(a ∪ b) ≤ ((a ∪ c)⊥ →2 b)

Proof of Theorem oa4lem1

Step	Hyp	Ref	Expression
1		leo 150	. . . . 5 a ≤ (a ∪ c)
2		ax-a1 29	. . . . 5 (a ∪ c) = (a ∪ c)⊥ ⊥
3	1, 2	lbtr 131	. . . 4 a ≤ (a ∪ c)⊥ ⊥
4		oa4lem1.1	. . . 4 a ≤ b⊥
5	3, 4	ler2an 165	. . 3 a ≤ ((a ∪ c)⊥ ⊥ ∩ b⊥ )
6	5	lelor 158	. 2 (b ∪ a) ≤ (b ∪ ((a ∪ c)⊥ ⊥ ∩ b⊥ ))
7		ax-a2 30	. 2 (a ∪ b) = (b ∪ a)
8		df-i2 44	. 2 ((a ∪ c)⊥ →2 b) = (b ∪ ((a ∪ c)⊥ ⊥ ∩ b⊥ ))
9	6, 7, 8	le3tr1 132	1 (a ∪ b) ≤ ((a ∪ c)⊥ →2 b)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14

This theorem is referenced by:  oa4lem3 919

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123

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