Theorem i3le 497

Description: L.e. to Kalmbach implication.

Hypothesis

Ref	Expression
i3le.1	(a →3 b) = 1

Assertion

Ref	Expression
i3le	a ≤ b

Proof of Theorem i3le

Step	Hyp	Ref	Expression
1		ancom 68	. . . 4 (1 ∩ b⊥ ) = (b⊥ ∩ 1)
2		i3le.1	. . . . . 6 (a →3 b) = 1
3	2	i3lem3 488	. . . . 5 ((a⊥ ∪ b) ∩ b⊥ ) = (a⊥ ∩ b⊥ )
4	2	i3lem4 489	. . . . . 6 (a⊥ ∪ b) = 1
5	4	ran 71	. . . . 5 ((a⊥ ∪ b) ∩ b⊥ ) = (1 ∩ b⊥ )
6		ancom 68	. . . . 5 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
7	3, 5, 6	3tr2 61	. . . 4 (1 ∩ b⊥ ) = (b⊥ ∩ a⊥ )
8		an1 98	. . . 4 (b⊥ ∩ 1) = b⊥
9	1, 7, 8	3tr2 61	. . 3 (b⊥ ∩ a⊥ ) = b⊥
10	9	df2le1 127	. 2 b⊥ ≤ a⊥
11	10	lecon1 147	1 a ≤ b

Syntax hints:   = wb 1   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₃ wi3 15

This theorem is referenced by:  binr1 499  binr2 500  binr3 501  i3ri3 520  i3li3 521  i32i3 522  u3lemle2 699

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-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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