Theorem u1lemana 587

Description: Lemma for Sasaki implication study.

Assertion

Ref	Expression
u1lemana	((a →1 b) ∩ a⊥ ) = a⊥

Proof of Theorem u1lemana

Step	Hyp	Ref	Expression
1		df-i1 43	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
2	1	ran 71	. 2 ((a →1 b) ∩ a⊥ ) = ((a⊥ ∪ (a ∩ b)) ∩ a⊥ )
3		ancom 68	. . 3 ((a⊥ ∪ (a ∩ b)) ∩ a⊥ ) = (a⊥ ∩ (a⊥ ∪ (a ∩ b)))
4		a5c 113	. . 3 (a⊥ ∩ (a⊥ ∪ (a ∩ b))) = a⊥
5	3, 4	ax-r2 35	. 2 ((a⊥ ∪ (a ∩ b)) ∩ a⊥ ) = a⊥
6	2, 5	ax-r2 35	1 ((a →1 b) ∩ a⊥ ) = a⊥

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13

This theorem is referenced by:  u1lemnoa 642  u12lembi 708  u1lem7 754

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-a 39  df-i1 43

metamath.org