Theorem womle 298

Description: An equality implying the WOM law. (Contributed by NM, 24-Jan-1999.)

Hypothesis

Ref	Expression
womle.1	(a ∩ (a →1 b)) = (a ∩ (a →2 b))

Assertion

Ref	Expression
womle	((a →2 b)⊥ ∪ (a →1 b)) = 1

Proof of Theorem womle

Step	Hyp	Ref	Expression
1		womle.1	. . . . 5 (a ∩ (a →1 b)) = (a ∩ (a →2 b))
2	1	ax-r1 35	. . . 4 (a ∩ (a →2 b)) = (a ∩ (a →1 b))
3		lear 161	. . . 4 (a ∩ (a →1 b)) ≤ (a →1 b)
4	2, 3	bltr 138	. . 3 (a ∩ (a →2 b)) ≤ (a →1 b)
5		leor 159	. . 3 (a →1 b) ≤ ((a →2 b)⊥ ∪ (a →1 b))
6	4, 5	letr 137	. 2 (a ∩ (a →2 b)) ≤ ((a →2 b)⊥ ∪ (a →1 b))
7	6	womle2a 295	1 ((a →2 b)⊥ ∪ (a →1 b)) = 1

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12   →₂ wi2 13

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

This theorem is used by: (None)