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Type | Label | Description |
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Statement | ||
Theorem | lub0N 32801 | The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.) |
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Theorem | opltn0 32802 | A lattice element greater than zero is nonzero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) |
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Theorem | ople1 32803 | Any element is less than the orthoposet unit. (chss 26938 analog.) (Contributed by NM, 23-Oct-2011.) |
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Theorem | op1le 32804 | If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 27152 analog.) (Contributed by NM, 5-Dec-2011.) |
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Theorem | glb0N 32805 | The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.) |
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Theorem | opoccl 32806 | Closure of orthocomplement operation. (choccl 27015 analog.) (Contributed by NM, 20-Oct-2011.) |
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Theorem | opococ 32807 | Double negative law for orthoposets. (ococ 27115 analog.) (Contributed by NM, 13-Sep-2011.) |
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Theorem | opcon3b 32808 | Contraposition law for orthoposets. (chcon3i 27175 analog.) (Contributed by NM, 8-Nov-2011.) |
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Theorem | opcon2b 32809 | Orthocomplement contraposition law. (negcon2 9958 analog.) (Contributed by NM, 16-Jan-2012.) |
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Theorem | opcon1b 32810 | Orthocomplement contraposition law. (negcon1 9957 analog.) (Contributed by NM, 24-Jan-2012.) |
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Theorem | oplecon3 32811 | Contraposition law for orthoposets. (Contributed by NM, 13-Sep-2011.) |
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Theorem | oplecon3b 32812 | Contraposition law for orthoposets. (chsscon3 27209 analog.) (Contributed by NM, 4-Nov-2011.) |
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Theorem | oplecon1b 32813 | Contraposition law for strict ordering in orthoposets. (chsscon1 27210 analog.) (Contributed by NM, 6-Nov-2011.) |
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Theorem | opoc1 32814 | Orthocomplement of orthoposet unit. (Contributed by NM, 24-Jan-2012.) |
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Theorem | opoc0 32815 | Orthocomplement of orthoposet zero. (Contributed by NM, 24-Jan-2012.) |
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Theorem | opltcon3b 32816 | Contraposition law for strict ordering in orthoposets. (chpsscon3 27212 analog.) (Contributed by NM, 4-Nov-2011.) |
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Theorem | opltcon1b 32817 | Contraposition law for strict ordering in orthoposets. (chpsscon1 27213 analog.) (Contributed by NM, 5-Nov-2011.) |
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Theorem | opltcon2b 32818 | Contraposition law for strict ordering in orthoposets. (chsscon2 27211 analog.) (Contributed by NM, 5-Nov-2011.) |
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Theorem | opexmid 32819 | Law of excluded middle for orthoposets. (chjo 27224 analog.) (Contributed by NM, 13-Sep-2011.) |
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Theorem | opnoncon 32820 | Law of contradiction for orthoposets. (chocin 27204 analog.) (Contributed by NM, 13-Sep-2011.) |
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Theorem | riotaocN 32821* |
The orthocomplement of the unique poset element such that ![]() |
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Theorem | cmtfvalN 32822* | Value of commutes relation. (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmtvalN 32823 | Equivalence for commutes relation. Definition of commutes in [Kalmbach] p. 20. (cmbr 27293 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.) |
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Theorem | isolat 32824 | The predicate "is an ortholattice." (Contributed by NM, 18-Sep-2011.) |
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Theorem | ollat 32825 | An ortholattice is a lattice. (Contributed by NM, 18-Sep-2011.) |
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Theorem | olop 32826 | An ortholattice is an orthoposet. (Contributed by NM, 18-Sep-2011.) |
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Theorem | olposN 32827 | An ortholattice is a poset. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.) |
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Theorem | isolatiN 32828 | Properties that determine an ortholattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.) |
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Theorem | oldmm1 32829 | De Morgan's law for meet in an ortholattice. (chdmm1 27234 analog.) (Contributed by NM, 6-Nov-2011.) |
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Theorem | oldmm2 32830 | De Morgan's law for meet in an ortholattice. (chdmm2 27235 analog.) (Contributed by NM, 6-Nov-2011.) |
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Theorem | oldmm3N 32831 | De Morgan's law for meet in an ortholattice. (chdmm3 27236 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | oldmm4 32832 | De Morgan's law for meet in an ortholattice. (chdmm4 27237 analog.) (Contributed by NM, 6-Nov-2011.) |
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Theorem | oldmj1 32833 | De Morgan's law for join in an ortholattice. (chdmj1 27238 analog.) (Contributed by NM, 6-Nov-2011.) |
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Theorem | oldmj2 32834 | De Morgan's law for join in an ortholattice. (chdmj2 27239 analog.) (Contributed by NM, 7-Nov-2011.) |
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Theorem | oldmj3 32835 | De Morgan's law for join in an ortholattice. (chdmj3 27240 analog.) (Contributed by NM, 7-Nov-2011.) |
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Theorem | oldmj4 32836 | De Morgan's law for join in an ortholattice. (chdmj4 27241 analog.) (Contributed by NM, 7-Nov-2011.) |
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Theorem | olj01 32837 | An ortholattice element joined with zero equals itself. (chj0 27206 analog.) (Contributed by NM, 19-Oct-2011.) |
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Theorem | olj02 32838 | An ortholattice element joined with zero equals itself. (Contributed by NM, 28-Jan-2012.) |
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Theorem | olm11 32839 | The meet of an ortholattice element with one equals itself. (chm1i 27165 analog.) (Contributed by NM, 22-May-2012.) |
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Theorem | olm12 32840 | The meet of an ortholattice element with one equals itself. (Contributed by NM, 22-May-2012.) |
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Theorem | latmassOLD 32841 | Ortholattice meet is associative. (This can also be proved for lattices with a longer proof.) (inass 3654 analog.) (Contributed by NM, 7-Nov-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | latm12 32842 | A rearrangement of lattice meet. (in12 3655 analog.) (Contributed by NM, 8-Nov-2011.) |
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Theorem | latm32 32843 | A rearrangement of lattice meet. (in12 3655 analog.) (Contributed by NM, 13-Nov-2012.) |
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Theorem | latmrot 32844 | Rotate lattice meet of 3 classes. (Contributed by NM, 9-Oct-2012.) |
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Theorem | latm4 32845 | Rearrangement of lattice meet of 4 classes. (in4 3660 analog.) (Contributed by NM, 8-Nov-2011.) |
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Theorem | latmmdiN 32846 | Lattice meet distributes over itself. (inindi 3661 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | latmmdir 32847 | Lattice meet distributes over itself. (inindir 3662 analog.) (Contributed by NM, 6-Jun-2012.) |
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Theorem | olm01 32848 | Meet with lattice zero is zero. (chm0 27200 analog.) (Contributed by NM, 8-Nov-2011.) |
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Theorem | olm02 32849 | Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.) |
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Theorem | isoml 32850* | The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.) |
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Theorem | isomliN 32851* | Properties that determine an orthomodular lattice. (Contributed by NM, 18-Sep-2011.) (New usage is discouraged.) |
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Theorem | omlol 32852 | An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.) |
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Theorem | omlop 32853 | An orthomodular lattice is an orthoposet. (Contributed by NM, 6-Nov-2011.) |
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Theorem | omllat 32854 | An orthomodular lattice is a lattice. (Contributed by NM, 6-Nov-2011.) |
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Theorem | omllaw 32855 | The orthomodular law. (Contributed by NM, 18-Sep-2011.) |
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Theorem | omllaw2N 32856 | Variation of orthomodular law. Definition of OML law in [Kalmbach] p. 22. (pjoml2i 27294 analog.) (Contributed by NM, 6-Nov-2011.) (New usage is discouraged.) |
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Theorem | omllaw3 32857 | Orthomodular law equivalent. Theorem 2(ii) of [Kalmbach] p. 22. (pjoml 27145 analog.) (Contributed by NM, 19-Oct-2011.) |
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Theorem | omllaw4 32858 | Orthomodular law equivalent. Remark in [Holland95] p. 223. (Contributed by NM, 19-Oct-2011.) |
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Theorem | omllaw5N 32859 | The orthomodular law. Remark in [Kalmbach] p. 22. (pjoml5 27322 analog.) (Contributed by NM, 14-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmtcomlemN 32860 | Lemma for cmtcomN 32861. (cmcmlem 27300 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmtcomN 32861 | Commutation is symmetric. Theorem 2(v) in [Kalmbach] p. 22. (cmcmi 27301 analog.) (Contributed by NM, 7-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmt2N 32862 | Commutation with orthocomplement. Theorem 2.3(i) of [Beran] p. 39. (cmcm2i 27302 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmt3N 32863 | Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27304 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmt4N 32864 | Commutation with orthocomplement. Remark in [Kalmbach] p. 23. (cmcm4i 27304 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmtbr2N 32865 | Alternate definition of the commutes relation. Remark in [Kalmbach] p. 23. (cmbr2i 27305 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmtbr3N 32866 | Alternate definition for the commutes relation. Lemma 3 of [Kalmbach] p. 23. (cmbr3 27317 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmtbr4N 32867 | Alternate definition for the commutes relation. (cmbr4i 27310 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.) |
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Theorem | lecmtN 32868 | Ordered elements commute. (lecmi 27311 analog.) (Contributed by NM, 10-Nov-2011.) (New usage is discouraged.) |
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Theorem | cmtidN 32869 | Any element commutes with itself. (cmidi 27319 analog.) (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.) |
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Theorem | omlfh1N 32870 | Foulis-Holland Theorem, part 1. If any 2 pairs in a triple of orthomodular lattice elements commute, the triple is distributive. Part of Theorem 5 in [Kalmbach] p. 25. (fh1 27327 analog.) (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | omlfh3N 32871 | Foulis-Holland Theorem, part 3. Dual of omlfh1N 32870. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.) |
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Theorem | omlmod1i2N 32872 | Analogue of modular law atmod1i2 33470 that holds in any OML. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.) |
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Theorem | omlspjN 32873 | Contraction of a Sasaki projection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.) |
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Syntax | ccvr 32874 | Extend class notation with covers relation. |
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Syntax | catm 32875 | Extend class notation with atoms. |
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Syntax | cal 32876 | Extend class notation with atomic lattices. |
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Syntax | clc 32877 | Extend class notation with lattices with the covering property. |
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Definition | df-covers 32878* |
Define the covers relation ("is covered by") for posets. "![]() ![]() ![]() ![]() |
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Definition | df-ats 32879* | Define the class of poset atoms. (Contributed by NM, 18-Sep-2011.) |
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Theorem | cvrfval 32880* | Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.) |
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Theorem | cvrval 32881* |
Binary relation expressing ![]() ![]() ![]() ![]() |
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Theorem | cvrlt 32882 | The covers relation implies the less-than relation. (cvpss 27994 analog.) (Contributed by NM, 8-Oct-2011.) |
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Theorem | cvrnbtwn 32883 | There is no element between the two arguments of the covers relation. (cvnbtwn 27995 analog.) (Contributed by NM, 18-Oct-2011.) |
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Theorem | ncvr1 32884 | No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.) |
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Theorem | cvrletrN 32885 | Property of an element above a covering. (Contributed by NM, 7-Dec-2012.) (New usage is discouraged.) |
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Theorem | cvrval2 32886* |
Binary relation expressing ![]() ![]() |
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Theorem | cvrnbtwn2 32887 | The covers relation implies no in-betweenness. (cvnbtwn2 27996 analog.) (Contributed by NM, 17-Nov-2011.) |
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Theorem | cvrnbtwn3 32888 | The covers relation implies no in-betweenness. (cvnbtwn3 27997 analog.) (Contributed by NM, 4-Nov-2011.) |
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Theorem | cvrcon3b 32889 | Contraposition law for the covers relation. (cvcon3 27993 analog.) (Contributed by NM, 4-Nov-2011.) |
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Theorem | cvrle 32890 | The covers relation implies the less-than-or-equal relation. (Contributed by NM, 12-Oct-2011.) |
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Theorem | cvrnbtwn4 32891 | The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 27998 analog.) (Contributed by NM, 18-Oct-2011.) |
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Theorem | cvrnle 32892 | The covers relation implies the negation of the reverse less-than-or-equal relation. (Contributed by NM, 18-Oct-2011.) |
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Theorem | cvrne 32893 | The covers relation implies inequality. (Contributed by NM, 13-Oct-2011.) |
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Theorem | cvrnrefN 32894 | The covers relation is not reflexive. (cvnref 28000 analog.) (Contributed by NM, 1-Nov-2012.) (New usage is discouraged.) |
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Theorem | cvrcmp 32895 | If two lattice elements that cover a third are comparable, then they are equal. (Contributed by NM, 6-Feb-2012.) |
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Theorem | cvrcmp2 32896 | If two lattice elements covered by a third are comparable, then they are equal. (Contributed by NM, 20-Jun-2012.) |
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Theorem | pats 32897* | The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.) |
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Theorem | isat 32898 | The predicate "is an atom". (ela 28048 analog.) (Contributed by NM, 18-Sep-2011.) |
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Theorem | isat2 32899 | The predicate "is an atom". (elatcv0 28050 analog.) (Contributed by NM, 18-Jun-2012.) |
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Theorem | atcvr0 32900 | An atom covers zero. (atcv0 28051 analog.) (Contributed by NM, 4-Nov-2011.) |
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