Theorem i3orlem4 555

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

Assertion

Ref	Expression
i3orlem4	((a ∪ c)⊥ ∩ (b ∪ c)) ≤ ((a ∪ c) →3 (b ∪ c))

Proof of Theorem i3orlem4

Step	Hyp	Ref	Expression
1		leo 158	. . 3 ((a ∪ c)⊥ ∩ (b ∪ c)) ≤ (((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ ))
2	1	ler 149	. 2 ((a ∪ c)⊥ ∩ (b ∪ c)) ≤ ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
3		df-i3 46	. . 3 ((a ∪ c) →3 (b ∪ c)) = ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))))
4	3	ax-r1 35	. 2 ((((a ∪ c)⊥ ∩ (b ∪ c)) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) ∪ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))) = ((a ∪ c) →3 (b ∪ c))
5	2, 4	lbtr 139	1 ((a ∪ c)⊥ ∩ (b ∪ c)) ≤ ((a ∪ c) →3 (b ∪ c))

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131

This theorem is used by: (None)