Theorem u1lemle2 697

Description: Sasaki implication to l.e.

Hypothesis

Ref	Expression
u1lemle2.1	(a ->1 b) = 1

Assertion

Ref	Expression
u1lemle2	a =< b

Proof of Theorem u1lemle2

Step	Hyp	Ref	Expression
1		anidm 103	. . . . . . . . 9 (a ^ a) = a
2	1	ran 71	. . . . . . . 8 ((a ^ a) ^ b) = (a ^ b)
3	2	ax-r1 34	. . . . . . 7 (a ^ b) = ((a ^ a) ^ b)
4		anass 69	. . . . . . 7 ((a ^ a) ^ b) = (a ^ (a ^ b))
5	3, 4	ax-r2 35	. . . . . 6 (a ^ b) = (a ^ (a ^ b))
6		dff 93	. . . . . 6 0 = (a ^ a_|_)
7	5, 6	2or 67	. . . . 5 ((a ^ b) v 0) = ((a ^ (a ^ b)) v (a ^ a_|_))
8		ax-a2 30	. . . . . . . 8 (a_|_ v (a ^ b)) = ((a ^ b) v a_|_)
9	8	lan 70	. . . . . . 7 (a ^ (a_|_ v (a ^ b))) = (a ^ ((a ^ b) v a_|_))
10		coman1 177	. . . . . . . 8 (a ^ b) C a
11	10	comcom2 175	. . . . . . . 8 (a ^ b) C a_|_
12	10, 11	fh2 452	. . . . . . 7 (a ^ ((a ^ b) v a_|_)) = ((a ^ (a ^ b)) v (a ^ a_|_))
13	9, 12	ax-r2 35	. . . . . 6 (a ^ (a_|_ v (a ^ b))) = ((a ^ (a ^ b)) v (a ^ a_|_))
14	13	ax-r1 34	. . . . 5 ((a ^ (a ^ b)) v (a ^ a_|_)) = (a ^ (a_|_ v (a ^ b)))
15	7, 14	ax-r2 35	. . . 4 ((a ^ b) v 0) = (a ^ (a_|_ v (a ^ b)))
16		df-i1 43	. . . . . . 7 (a ->1 b) = (a_|_ v (a ^ b))
17	16	ax-r1 34	. . . . . 6 (a_|_ v (a ^ b)) = (a ->1 b)
18		u1lemle2.1	. . . . . 6 (a ->1 b) = 1
19	17, 18	ax-r2 35	. . . . 5 (a_|_ v (a ^ b)) = 1
20	19	lan 70	. . . 4 (a ^ (a_|_ v (a ^ b))) = (a ^ 1)
21	15, 20	ax-r2 35	. . 3 ((a ^ b) v 0) = (a ^ 1)
22		or0 94	. . 3 ((a ^ b) v 0) = (a ^ b)
23		an1 98	. . 3 (a ^ 1) = a
24	21, 22, 23	3tr2 61	. 2 (a ^ b) = a
25	24	df2le1 127	1 a =< b

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->1 wi1 13

This theorem is referenced by:  3vded11 796  3vded12 797

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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