Theorem wlem3.1 202

Description: Weak analogue to lemma used in proof of Th. 3.1 of Pavicic 1993.

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 86	. . 3 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2		wlem3.1.1	. . . . . 6 (a ∪ b) = b
3	2	leoa 115	. . . . 5 (a ∩ b) = a
4		oran 79	. . . . . . . 8 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
5	4	ax-r1 34	. . . . . . 7 (a⊥ ∩ b⊥ )⊥ = (a ∪ b)
6	5, 2	ax-r2 35	. . . . . 6 (a⊥ ∩ b⊥ )⊥ = b
7	6	con3 65	. . . . 5 (a⊥ ∩ b⊥ ) = b⊥
8	3, 7	2or 67	. . . 4 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (a ∪ b⊥ )
9		ax-a2 30	. . . 4 (a ∪ b⊥ ) = (b⊥ ∪ a)
10	8, 9	ax-r2 35	. . 3 ((a ∩ b) ∪ (a⊥ ∩ b⊥ )) = (b⊥ ∪ a)
11	1, 10	ax-r2 35	. 2 (a ≡ b) = (b⊥ ∪ a)
12		wlem3.1.2	. 2 (b⊥ ∪ a) = 1
13	11, 12	ax-r2 35	1 (a ≡ b) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9

This theorem is referenced by:  woml 203

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-b 38  df-a 39

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