Theorem 2atneat

Description: The join of two distinct atoms is not an atom. (Contributed by NM, 12-Oct-2012)

	Ref	Expression
Hypotheses	2atneat.j	|- .\/ = ( join ` K )
	2atneat.a	|- A = ( Atoms ` K )
Assertion	2atneat	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> -. ( P .\/ Q ) e. A )

Proof

Step	Hyp	Ref	Expression
1		2atneat.j	|- .\/ = ( join ` K )
2		2atneat.a	|- A = ( Atoms ` K )
3		simpl	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> K e. HL )
4		simpr1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P e. A )
5		simpr2	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> Q e. A )
6		simpr3	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P =/= Q )
7		eqid	|- ( LLines ` K ) = ( LLines ` K )
8	1 2 7	llni2	|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( LLines ` K ) )
9	3 4 5 6 8	syl31anc	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( P .\/ Q ) e. ( LLines ` K ) )
10	2 7	llnneat	|- ( ( K e. HL /\ ( P .\/ Q ) e. ( LLines ` K ) ) -> -. ( P .\/ Q ) e. A )
11	9 10	syldan	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> -. ( P .\/ Q ) e. A )