Description: Part of proof of Lemma E in Crawley p. 113. C represents s_1, which we prove is an atom. (Contributed by NM, 10-Jun-2012)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cdleme8.l | |- .<_ = ( le ` K ) |
|
| cdleme8.j | |- .\/ = ( join ` K ) |
||
| cdleme8.m | |- ./\ = ( meet ` K ) |
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| cdleme8.a | |- A = ( Atoms ` K ) |
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| cdleme8.h | |- H = ( LHyp ` K ) |
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| cdleme8.4 | |- C = ( ( P .\/ S ) ./\ W ) |
||
| Assertion | cdleme9a | |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ P =/= S ) ) -> C e. A ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdleme8.l | |- .<_ = ( le ` K ) |
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| 2 | cdleme8.j | |- .\/ = ( join ` K ) |
|
| 3 | cdleme8.m | |- ./\ = ( meet ` K ) |
|
| 4 | cdleme8.a | |- A = ( Atoms ` K ) |
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| 5 | cdleme8.h | |- H = ( LHyp ` K ) |
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| 6 | cdleme8.4 | |- C = ( ( P .\/ S ) ./\ W ) |
|
| 7 | 1 2 3 4 5 6 | lhpat2 | |- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ P =/= S ) ) -> C e. A ) |