Theorem wlecon 395

Description: Contrapositive for l.e. (Contributed by NM, 13-Oct-1997.)

Hypothesis

Ref	Expression
wle.1	(a ≤2 b) = 1

Assertion

Ref	Expression
wlecon	(b⊥ ≤2 a⊥ ) = 1

Proof of Theorem wlecon

Step	Hyp	Ref	Expression
1		ax-a2 31	. . . . 5 (b ∪ a) = (a ∪ b)
2	1	bi1 118	. . . 4 ((b ∪ a) ≡ (a ∪ b)) = 1
3		oran 87	. . . . 5 (b ∪ a) = (b⊥ ∩ a⊥ )⊥
4	3	bi1 118	. . . 4 ((b ∪ a) ≡ (b⊥ ∩ a⊥ )⊥ ) = 1
5		wle.1	. . . . 5 (a ≤2 b) = 1
6	5	wdf-le2 379	. . . 4 ((a ∪ b) ≡ b) = 1
7	2, 4, 6	w3tr2 375	. . 3 ((b⊥ ∩ a⊥ )⊥ ≡ b) = 1
8	7	wcon3 209	. 2 ((b⊥ ∩ a⊥ ) ≡ b⊥ ) = 1
9	8	wdf2le1 385	1 (b⊥ ≤2 a⊥ ) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   ≤₂ wle2 10

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131

This theorem is referenced by: (None)