Theorem wleao 377

Description: Relation between two methods of expressing "less than or equal to". (Contributed by NM, 27-Sep-1997.)

Hypothesis

Ref	Expression
wleao.1	((c ∩ b) ≡ a) = 1

Assertion

Ref	Expression
wleao	((a ∪ b) ≡ b) = 1

Proof of Theorem wleao

Step	Hyp	Ref	Expression
1		wa2 192	. . 3 ((a ∪ b) ≡ (b ∪ a)) = 1
2		wleao.1	. . . . . 6 ((c ∩ b) ≡ a) = 1
3	2	wr1 197	. . . . 5 (a ≡ (c ∩ b)) = 1
4		wancom 203	. . . . . 6 ((b ∩ c) ≡ (c ∩ b)) = 1
5	4	wr1 197	. . . . 5 ((c ∩ b) ≡ (b ∩ c)) = 1
6	3, 5	wr2 371	. . . 4 (a ≡ (b ∩ c)) = 1
7	6	wlor 368	. . 3 ((b ∪ a) ≡ (b ∪ (b ∩ c))) = 1
8	1, 7	wr2 371	. 2 ((a ∪ b) ≡ (b ∪ (b ∩ c))) = 1
9		wa5b 200	. 2 ((b ∪ (b ∩ c)) ≡ b) = 1
10	8, 9	wr2 371	1 ((a ∪ b) ≡ b) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8

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-le1 130  df-le2 131

This theorem is referenced by:  wdf2le1  385