Theorem wleao 359

Description: Relation between two methods of expressing "less than or equal to".

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 184	. . 3 ((a ∪ b) ≡ (b ∪ a)) = 1
2		wleao.1	. . . . . 6 ((c ∩ b) ≡ a) = 1
3	2	wr1 189	. . . . 5 (a ≡ (c ∩ b)) = 1
4		wancom 195	. . . . . 6 ((b ∩ c) ≡ (c ∩ b)) = 1
5	4	wr1 189	. . . . 5 ((c ∩ b) ≡ (b ∩ c)) = 1
6	3, 5	wr2 353	. . . 4 (a ≡ (b ∩ c)) = 1
7	6	wlor 350	. . 3 ((b ∪ a) ≡ (b ∪ (b ∩ c))) = 1
8	1, 7	wr2 353	. 2 ((a ∪ b) ≡ (b ∪ (b ∩ c))) = 1
9		wa5b 192	. 2 ((b ∪ (b ∩ c)) ≡ b) = 1
10	8, 9	wr2 353	1 ((a ∪ b) ≡ b) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9

This theorem is referenced by:  wdf2le1 367

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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