Theorem i3orlem2 553

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

Assertion

Ref	Expression
i3orlem2	(a ∩ b) ≤ ((a ∪ c) →3 (b ∪ c))

Proof of Theorem i3orlem2

Step	Hyp	Ref	Expression
1		leo 158	. . 3 a ≤ (a ∪ c)
2		leo 158	. . 3 b ≤ (b ∪ c)
3	1, 2	le2an 169	. 2 (a ∩ b) ≤ ((a ∪ c) ∩ (b ∪ c))
4		leor 159	. . . 4 ((a ∪ c) ∩ (b ∪ c)) ≤ (((a ∪ c) ∩ (a ∪ c)⊥ ) ∪ ((a ∪ c) ∩ (b ∪ c)))
5		ledi 174	. . . 4 (((a ∪ c) ∩ (a ∪ c)⊥ ) ∪ ((a ∪ c) ∩ (b ∪ c))) ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))
6	4, 5	letr 137	. . 3 ((a ∪ c) ∩ (b ∪ c)) ≤ ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c)))
7		i3orlem1 552	. . 3 ((a ∪ c) ∩ ((a ∪ c)⊥ ∪ (b ∪ c))) ≤ ((a ∪ c) →3 (b ∪ c))
8	6, 7	letr 137	. 2 ((a ∪ c) ∩ (b ∪ c)) ≤ ((a ∪ c) →3 (b ∪ c))
9	3, 8	letr 137	1 (a ∩ b) ≤ ((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-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131

This theorem is used by:  i3orlem6  557