Theorem oa4lem3 939

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

Hypotheses

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

Assertion

Ref	Expression
oa4lem3	((a ∪ b) ∩ (c ∪ d)) ≤ ((b ∪ d)⊥ ∪ (((a ∪ c)⊥ →2 b) ∩ ((a ∪ c)⊥ →2 d)))

Proof of Theorem oa4lem3

Step	Hyp	Ref	Expression
1		oa4lem1.1	. . . 4 a ≤ b⊥
2		oa4lem1.2	. . . 4 c ≤ d⊥
3	1, 2	oa4lem1 937	. . 3 (a ∪ b) ≤ ((a ∪ c)⊥ →2 b)
4	1, 2	oa4lem2 938	. . 3 (c ∪ d) ≤ ((a ∪ c)⊥ →2 d)
5	3, 4	le2an 169	. 2 ((a ∪ b) ∩ (c ∪ d)) ≤ (((a ∪ c)⊥ →2 b) ∩ ((a ∪ c)⊥ →2 d))
6		leor 159	. 2 (((a ∪ c)⊥ →2 b) ∩ ((a ∪ c)⊥ →2 d)) ≤ ((b ∪ d)⊥ ∪ (((a ∪ c)⊥ →2 b) ∩ ((a ∪ c)⊥ →2 d)))
7	5, 6	letr 137	1 ((a ∪ b) ∩ (c ∪ d)) ≤ ((b ∪ d)⊥ ∪ (((a ∪ c)⊥ →2 b) ∩ ((a ∪ c)⊥ →2 d)))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

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-i2 45  df-le1 130  df-le2 131

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