Theorem u2lemle2 698

Description: Dishkant implication to l.e.

Hypothesis

Ref	Expression
u2lemle2.1	(a ->2 b) = 1

Assertion

Ref	Expression
u2lemle2	a =< b

Proof of Theorem u2lemle2

Step	Hyp	Ref	Expression
1		ax-a2 30	. . . . . . 7 (b v (a_|_ ^ b_|_)) = ((a_|_ ^ b_|_) v b)
2	1	lan 70	. . . . . 6 (a ^ (b v (a_|_ ^ b_|_))) = (a ^ ((a_|_ ^ b_|_) v b))
3		coman1 177	. . . . . . . . 9 (a_|_ ^ b_|_) C a_|_
4	3	comcom7 442	. . . . . . . 8 (a_|_ ^ b_|_) C a
5		coman2 178	. . . . . . . . 9 (a_|_ ^ b_|_) C b_|_
6	5	comcom7 442	. . . . . . . 8 (a_|_ ^ b_|_) C b
7	4, 6	fh2 452	. . . . . . 7 (a ^ ((a_|_ ^ b_|_) v b)) = ((a ^ (a_|_ ^ b_|_)) v (a ^ b))
8		ancom 68	. . . . . . . . . 10 ((a ^ a_|_) ^ b_|_) = (b_|_ ^ (a ^ a_|_))
9		anass 69	. . . . . . . . . 10 ((a ^ a_|_) ^ b_|_) = (a ^ (a_|_ ^ b_|_))
10		dff 93	. . . . . . . . . . . . 13 0 = (a ^ a_|_)
11	10	ax-r1 34	. . . . . . . . . . . 12 (a ^ a_|_) = 0
12	11	lan 70	. . . . . . . . . . 11 (b_|_ ^ (a ^ a_|_)) = (b_|_ ^ 0)
13		an0 100	. . . . . . . . . . 11 (b_|_ ^ 0) = 0
14	12, 13	ax-r2 35	. . . . . . . . . 10 (b_|_ ^ (a ^ a_|_)) = 0
15	8, 9, 14	3tr2 61	. . . . . . . . 9 (a ^ (a_|_ ^ b_|_)) = 0
16	15	ax-r5 37	. . . . . . . 8 ((a ^ (a_|_ ^ b_|_)) v (a ^ b)) = (0 v (a ^ b))
17		ax-a2 30	. . . . . . . 8 (0 v (a ^ b)) = ((a ^ b) v 0)
18	16, 17	ax-r2 35	. . . . . . 7 ((a ^ (a_|_ ^ b_|_)) v (a ^ b)) = ((a ^ b) v 0)
19	7, 18	ax-r2 35	. . . . . 6 (a ^ ((a_|_ ^ b_|_) v b)) = ((a ^ b) v 0)
20	2, 19	ax-r2 35	. . . . 5 (a ^ (b v (a_|_ ^ b_|_))) = ((a ^ b) v 0)
21	20	ax-r1 34	. . . 4 ((a ^ b) v 0) = (a ^ (b v (a_|_ ^ b_|_)))
22		df-i2 44	. . . . . . 7 (a ->2 b) = (b v (a_|_ ^ b_|_))
23	22	ax-r1 34	. . . . . 6 (b v (a_|_ ^ b_|_)) = (a ->2 b)
24		u2lemle2.1	. . . . . 6 (a ->2 b) = 1
25	23, 24	ax-r2 35	. . . . 5 (b v (a_|_ ^ b_|_)) = 1
26	25	lan 70	. . . 4 (a ^ (b v (a_|_ ^ b_|_))) = (a ^ 1)
27	21, 26	ax-r2 35	. . 3 ((a ^ b) v 0) = (a ^ 1)
28		or0 94	. . 3 ((a ^ b) v 0) = (a ^ b)
29		an1 98	. . 3 (a ^ 1) = a
30	27, 28, 29	3tr2 61	. 2 (a ^ b) = a
31	30	df2le1 127	1 a =< b

Syntax hints:   = wb 1   =< wle 2  _|_wn 4   v wo 6   ^ wa 7  1wt 9  0wf 10   ->2 wi2 14

This theorem is referenced by:  3vroa 813  imp3 823

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-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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