Theorem i1com 708

Description: Commutation expressed with →₁ . (Contributed by NM, 1-Dec-1999.)

Hypothesis

Ref	Expression
i1com.1	b ≤ (a →1 b)

Assertion

Ref	Expression
i1com	a C b

Proof of Theorem i1com

Step	Hyp	Ref	Expression
1		ancom 74	. . . 4 (b ∩ (a →1 b)) = ((a →1 b) ∩ b)
2		i1com.1	. . . . 5 b ≤ (a →1 b)
3	2	df2le2 136	. . . 4 (b ∩ (a →1 b)) = b
4		u1lemab 610	. . . . 5 ((a →1 b) ∩ b) = ((a ∩ b) ∪ (a⊥ ∩ b))
5		ancom 74	. . . . . 6 (a ∩ b) = (b ∩ a)
6		ancom 74	. . . . . 6 (a⊥ ∩ b) = (b ∩ a⊥ )
7	5, 6	2or 72	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b)) = ((b ∩ a) ∪ (b ∩ a⊥ ))
8	4, 7	ax-r2 36	. . . 4 ((a →1 b) ∩ b) = ((b ∩ a) ∪ (b ∩ a⊥ ))
9	1, 3, 8	3tr2 64	. . 3 b = ((b ∩ a) ∪ (b ∩ a⊥ ))
10	9	df-c1 132	. 2 b C a
11	10	comcom 453	1 a C b

Syntax hints:   ≤ wle 2   C wc 3  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  comanb  872