Description: Lemma for 4atexlem7 . (Contributed by NM, 23-Nov-2012)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | 4thatlem.ph | |- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) ) |
|
| 4thatlemslps.l | |- .<_ = ( le ` K ) |
||
| 4thatlemslps.j | |- .\/ = ( join ` K ) |
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| 4thatlemslps.a | |- A = ( Atoms ` K ) |
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| Assertion | 4atexlempns | |- ( ph -> P =/= S ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 4thatlem.ph | |- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) ) |
|
| 2 | 4thatlemslps.l | |- .<_ = ( le ` K ) |
|
| 3 | 4thatlemslps.j | |- .\/ = ( join ` K ) |
|
| 4 | 4thatlemslps.a | |- A = ( Atoms ` K ) |
|
| 5 | 1 | 4atexlemk | |- ( ph -> K e. HL ) |
| 6 | 1 | 4atexlemp | |- ( ph -> P e. A ) |
| 7 | 1 | 4atexlemq | |- ( ph -> Q e. A ) |
| 8 | 1 | 4atexlems | |- ( ph -> S e. A ) |
| 9 | 1 | 4atexlemnslpq | |- ( ph -> -. S .<_ ( P .\/ Q ) ) |
| 10 | 2 3 4 | 4atlem0be | |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ S e. A ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= S ) |
| 11 | 5 6 7 8 9 10 | syl131anc | |- ( ph -> P =/= S ) |