Theorem i3orlem5 556

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

Assertion

Ref	Expression
i3orlem5	((a⊥ ∩ b⊥ ) ∩ c⊥ ) ≤ ((a ∪ c) →3 (b ∪ c))

Proof of Theorem i3orlem5

Step	Hyp	Ref	Expression
1		leo 158	. 2 ((a ∪ c)⊥ ∩ (b ∪ c)⊥ ) ≤ (((a ∪ c)⊥ ∩ (b ∪ c)⊥ ) ∪ (((a ∪ c)⊥ ∪ (b ∪ c)) ∩ ((a ∪ c) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)))))
2		anandir 115	. . 3 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) = ((a⊥ ∩ c⊥ ) ∩ (b⊥ ∩ c⊥ ))
3		oran 87	. . . . . 6 (a ∪ c) = (a⊥ ∩ c⊥ )⊥
4	3	con2 67	. . . . 5 (a ∪ c)⊥ = (a⊥ ∩ c⊥ )
5	4	ax-r1 35	. . . 4 (a⊥ ∩ c⊥ ) = (a ∪ c)⊥
6		oran 87	. . . . . 6 (b ∪ c) = (b⊥ ∩ c⊥ )⊥
7	6	con2 67	. . . . 5 (b ∪ c)⊥ = (b⊥ ∩ c⊥ )
8	7	ax-r1 35	. . . 4 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
9	5, 8	2an 79	. . 3 ((a⊥ ∩ c⊥ ) ∩ (b⊥ ∩ c⊥ )) = ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )
10	2, 9	ax-r2 36	. 2 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) = ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )
11		df2i3 498	. 2 ((a ∪ c) →3 (b ∪ c)) = (((a ∪ c)⊥ ∩ (b ∪ c)⊥ ) ∪ (((a ∪ c)⊥ ∪ (b ∪ c)) ∩ ((a ∪ c) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)))))
12	1, 10, 11	le3tr1 140	1 ((a⊥ ∩ 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-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)