Theorem lecon1 155

Description: Contrapositive for l.e. (Contributed by NM, 7-Nov-1997.)

Hypothesis

Ref	Expression
lecon1.1	a⊥ ≤ b⊥

Assertion

Ref	Expression
lecon1	b ≤ a

Proof of Theorem lecon1

Step	Hyp	Ref	Expression
1		lecon1.1	. . 3 a⊥ ≤ b⊥
2	1	lecon 154	. 2 b⊥ ⊥ ≤ a⊥ ⊥
3		ax-a1 30	. 2 b = b⊥ ⊥
4		ax-a1 30	. 2 a = a⊥ ⊥
5	2, 3, 4	le3tr1 140	1 b ≤ a

Syntax hints:   ≤ wle 2  ^⊥ wn 4

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-le1 130  df-le2 131

This theorem is referenced by:  lecon2  156  lecon3  157  i3le  515  neg3antlem2  865  elimcons  868  oa4v3v  934  oa3to4lem6  950  oa4uto4g  975  oa4uto4  977