Theorem wlem3.1 210

Description: Weak analogue to lemma used in proof of Thm. 3.1 of Pavicic 1993. (Contributed by NM, 2-Sep-1997.)

Hypotheses

Ref	Expression
wlem3.1.1	(a ∪ b) = b
wlem3.1.2	(b⊥ ∪ a) = 1

Assertion

Ref	Expression
wlem3.1	(a ≡ b) = 1

Proof of Theorem wlem3.1

Step	Hyp	Ref	Expression
1		dfb 94	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		wlem3.1.1	. . . . . 6 (a ∪ b) = b
3	2	leoa 123	. . . . 5 (a ∩ b) = a
4		oran 87	. . . . . . . 8 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
5	4	ax-r1 35	. . . . . . 7 (a⊥ ∩ b⊥ )⊥ = (a ∪ b)
6	5, 2	ax-r2 36	. . . . . 6 (a⊥ ∩ b⊥ )⊥ = b
7	6	con3 68	. . . . 5 (a⊥ ∩ b⊥ ) = b⊥
8	3, 7	2or 72	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ∪ b⊥ )
9		ax-a2 31	. . . 4 (a ∪ b⊥ ) = (b⊥ ∪ a)
10	8, 9	ax-r2 36	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (b⊥ ∪ a)
11	1, 10	ax-r2 36	. 2 (a ≡ b) = (b⊥ ∪ a)
12		wlem3.1.2	. 2 (b⊥ ∪ a) = 1
13	11, 12	ax-r2 36	1 (a ≡ b) = 1

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

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  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

This theorem is referenced by:  woml  211  lem3.1  443