Theorem i3lem2 505

Description: Lemma for Kalmbach implication. (Contributed by NM, 7-Nov-1997.)

Hypothesis

Ref	Expression
i3lem.1	(a →3 b) = 1

Assertion

Ref	Expression
i3lem2	a C b

Proof of Theorem i3lem2

Step	Hyp	Ref	Expression
1		i3lem.1	. . . . . 6 (a →3 b) = 1
2	1	i3lem1 504	. . . . 5 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = a⊥
3	2	ax-r1 35	. . . 4 a⊥ = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
4	3	df-c1 132	. . 3 a⊥ C b
5	4	comcom2 183	. 2 a⊥ C b⊥
6	5	comcom5 458	1 a C b

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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)