Theorem wlem1 243

Description: Lemma for 2-variable WOML proof. (Contributed by NM, 11-Nov-1998.)

Assertion

Ref	Expression
wlem1	((a ≡ b)⊥ ∪ ((a →1 b) ∩ (b →1 a))) = 1

Proof of Theorem wlem1

Step	Hyp	Ref	Expression
1		le1 146	. 2 ((a ≡ b)⊥ ∪ ((a →1 b) ∩ (b →1 a))) ≤ 1
2		df-t 41	. . . 4 1 = ((a ≡ b) ∪ (a ≡ b)⊥ )
3		ax-a2 31	. . . 4 ((a ≡ b) ∪ (a ≡ b)⊥ ) = ((a ≡ b)⊥ ∪ (a ≡ b))
4	2, 3	ax-r2 36	. . 3 1 = ((a ≡ b)⊥ ∪ (a ≡ b))
5		dfb 94	. . . . 5 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
6		ledio 176	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ (((a ∩ b) ∪ a⊥ ) ∩ ((a ∩ b) ∪ b⊥ ))
7		df-i1 44	. . . . . . . . 9 (a →1 b) = (a⊥ ∪ (a ∩ b))
8		ax-a2 31	. . . . . . . . 9 (a⊥ ∪ (a ∩ b)) = ((a ∩ b) ∪ a⊥ )
9	7, 8	ax-r2 36	. . . . . . . 8 (a →1 b) = ((a ∩ b) ∪ a⊥ )
10		df-i1 44	. . . . . . . . 9 (b →1 a) = (b⊥ ∪ (b ∩ a))
11		ax-a2 31	. . . . . . . . . 10 (b⊥ ∪ (b ∩ a)) = ((b ∩ a) ∪ b⊥ )
12		ancom 74	. . . . . . . . . . 11 (b ∩ a) = (a ∩ b)
13	12	ax-r5 38	. . . . . . . . . 10 ((b ∩ a) ∪ b⊥ ) = ((a ∩ b) ∪ b⊥ )
14	11, 13	ax-r2 36	. . . . . . . . 9 (b⊥ ∪ (b ∩ a)) = ((a ∩ b) ∪ b⊥ )
15	10, 14	ax-r2 36	. . . . . . . 8 (b →1 a) = ((a ∩ b) ∪ b⊥ )
16	9, 15	2an 79	. . . . . . 7 ((a →1 b) ∩ (b →1 a)) = (((a ∩ b) ∪ a⊥ ) ∩ ((a ∩ b) ∪ b⊥ ))
17	16	ax-r1 35	. . . . . 6 (((a ∩ b) ∪ a⊥ ) ∩ ((a ∩ b) ∪ b⊥ )) = ((a →1 b) ∩ (b →1 a))
18	6, 17	lbtr 139	. . . . 5 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) ≤ ((a →1 b) ∩ (b →1 a))
19	5, 18	bltr 138	. . . 4 (a ≡ b) ≤ ((a →1 b) ∩ (b →1 a))
20	19	lelor 166	. . 3 ((a ≡ b)⊥ ∪ (a ≡ b)) ≤ ((a ≡ b)⊥ ∪ ((a →1 b) ∩ (b →1 a)))
21	4, 20	bltr 138	. 2 1 ≤ ((a ≡ b)⊥ ∪ ((a →1 b) ∩ (b →1 a)))
22	1, 21	lebi 145	1 ((a ≡ b)⊥ ∪ ((a →1 b) ∩ (b →1 a))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

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

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

This theorem is referenced by:  wr5-2v  366