Theorem i3orlem1 552

Description: Lemma for Kalmbach implication OR builder. (Contributed by NM, 11-Nov-1997.)

Assertion

Ref	Expression
i3orlem1	((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))

Proof of Theorem i3orlem1

Step	Hyp	Ref	Expression
1		leor 159	. 2 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
2		df-i3 46	. . 3 ((a ∪ c) →3 (b ∪ c)) = ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
3	2	ax-r1 35	. 2 ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))) = ((a ∪ c) →3 (b ∪ c))
4	1, 3	lbtr 139	1 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i3 46  df-le1 130  df-le2 131

This theorem is referenced by:  i3orlem2  553  i3orlem3  554