Theorem i3orlem5 538

Description: Lemma for Kalmbach implication OR builder.

Assertion

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

Proof of Theorem i3orlem5

Step	Hyp	Ref	Expression
1		leo 150	. 2 ((a ∪ c)⊥ ∩ (b ∪ c)⊥ ) ≤ (((a ∪ c)⊥ ∩ (b ∪ c)⊥ ) ∪ (((a ∪ c)⊥ ∪ (b ∪ c)) ∩ ((a ∪ c) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)))))
2		anandir 107	. . 3 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) = ((a⊥ ∩ c⊥ ) ∩ (b⊥ ∩ c⊥ ))
3		oran 79	. . . . . 6 (a ∪ c) = (a⊥ ∩ c⊥ )⊥
4	3	con2 64	. . . . 5 (a ∪ c)⊥ = (a⊥ ∩ c⊥ )
5	4	ax-r1 34	. . . 4 (a⊥ ∩ c⊥ ) = (a ∪ c)⊥
6		oran 79	. . . . . 6 (b ∪ c) = (b⊥ ∩ c⊥ )⊥
7	6	con2 64	. . . . 5 (b ∪ c)⊥ = (b⊥ ∩ c⊥ )
8	7	ax-r1 34	. . . 4 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
9	5, 8	2an 72	. . 3 ((a⊥ ∩ c⊥ ) ∩ (b⊥ ∩ c⊥ )) = ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )
10	2, 9	ax-r2 35	. 2 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) = ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )
11		df2i3 480	. 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 132	1 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) ≤ ((a ∪ c) →3 (b ∪ c))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₃ wi3 15

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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