Metamath Proof Explorer


Theorem chpssati

Description: Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)

Ref Expression
Hypotheses chpssat.1 AC
chpssat.2 BC
Assertion chpssati ABxHAtomsxB¬xA

Proof

Step Hyp Ref Expression
1 chpssat.1 AC
2 chpssat.2 BC
3 dfpss3 ABAB¬BA
4 3 simprbi AB¬BA
5 iman xBxA¬xB¬xA
6 5 ralbii xHAtomsxBxAxHAtoms¬xB¬xA
7 ss2rab xHAtoms|xBxHAtoms|xAxHAtomsxBxA
8 ssrab2 xHAtoms|xBHAtoms
9 atssch HAtomsC
10 8 9 sstri xHAtoms|xBC
11 ssrab2 xHAtoms|xAHAtoms
12 11 9 sstri xHAtoms|xAC
13 chsupss xHAtoms|xBCxHAtoms|xACxHAtoms|xBxHAtoms|xAxHAtoms|xBxHAtoms|xA
14 10 12 13 mp2an xHAtoms|xBxHAtoms|xAxHAtoms|xBxHAtoms|xA
15 2 hatomistici B=xHAtoms|xB
16 1 hatomistici A=xHAtoms|xA
17 14 15 16 3sstr4g xHAtoms|xBxHAtoms|xABA
18 7 17 sylbir xHAtomsxBxABA
19 6 18 sylbir xHAtoms¬xB¬xABA
20 19 con3i ¬BA¬xHAtoms¬xB¬xA
21 dfrex2 xHAtomsxB¬xA¬xHAtoms¬xB¬xA
22 20 21 sylibr ¬BAxHAtomsxB¬xA
23 4 22 syl ABxHAtomsxB¬xA