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Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdomrefg 7101 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)

Theoremen2d 7102* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)

Theoremen3d 7103* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)

Theoremen2i 7104* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)

Theoremen3i 7105* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)

Theoremdom2lem 7106* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)

Theoremdom2d 7107* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 20-May-2013.)

Theoremdom3d 7108* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)

Theoremdom2 7109* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. and can be read and , as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)

Theoremdom3 7110* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. and can be read and , as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)

Theoremidssen 7111 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremssdomg 7112 A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremener 7113 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremensymb 7114 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremensym 7115 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremensymi 7116 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)

Theoremensymd 7117 Symmetry of equinumerosity. Deduction form of ensym 7115. (Contributed by David Moews, 1-May-2017.)

Theorementr 7118 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)

Theoremdomtr 7119 Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theorementri 7120 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)

Theorementr2i 7121 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)

Theorementr3i 7122 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)

Theorementr4i 7123 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)

Theoremendomtr 7124 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)

Theoremdomentr 7125 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)

Theoremf1imaeng 7126 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)

Theoremf1imaen2g 7127 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7128 does not need ax-reg 7516.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)

Theoremf1imaen 7128 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)

Theoremen0 7129 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)

Theoremensn1 7130 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)

Theoremensn1g 7131 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)

Theoremenpr1g 7132 has only one element. (Contributed by FL, 15-Feb-2010.)

Theoremen1 7133* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)

Theoremen1b 7134 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)

Theoremreuen1 7135* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremeuen1 7136 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)

Theoremeuen1b 7137* Two ways to express " has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)

Theorem2dom 7138* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)

Theoremfundmen 7139 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremfundmeng 7140 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)

Theoremcnven 7141 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)

Theoremfndmeng 7142 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremmapsnen 7143 Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremmap1 7144 Set exponentiation: ordinal 1 to any set is equinumerous to ordinal 1. Exercise 4.42(b) of [Mendelson] p. 255. (Contributed by NM, 17-Dec-2003.)

Theoremen2sn 7145 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)

Theoremsnfi 7146 A singleton is finite. (Contributed by NM, 4-Nov-2002.)

Theoremfiprc 7147 The class of finite sets is a proper class. (Contributed by Jeffrey Hankins, 3-Oct-2008.)

Theoremunen 7148 Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremdifsnen 7149 All decrements of a set are equinumerous. (Contributed by Stefan O'Rear, 19-Feb-2015.)

Theoremdomdifsn 7150 Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)

Theoremxpsnen 7151 A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremxpsneng 7152 A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)

Theoremxp1en 7153 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremendisj 7154* Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)

Theoremundom 7155 Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremxpcomf1o 7156* The canonical bijection from to . (Contributed by Mario Carneiro, 23-Apr-2014.)

Theoremxpcomco 7157* Composition with the bijection of xpcomf1o 7156 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)

Theoremxpcomen 7158 Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremxpcomeng 7159 Commutative law for equinumerosity of cross product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)

Theoremxpsnen2g 7160 A set is equinumerous to its cross-product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)

Theoremxpassen 7161 Associative law for equinumerosity of cross product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremxpdom2 7162 Dominance law for cross product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Theoremxpdom2g 7163 Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)

Theoremxpdom1g 7164 Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremxpdom3 7165 A set is dominated by its cross product with a non-empty set. Exercise 6 of [Suppes] p. 98. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Theoremxpdom1 7166 Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Mar-2006.)

Theoremdomunsncan 7167 A singleton cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)

Theoremomxpenlem 7168* Lemma for omxpen 7169. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 25-May-2015.)

Theoremomxpen 7169 The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.)

Theoremomf1o 7170* Construct an explicit bijection from to . (Contributed by Mario Carneiro, 30-May-2015.)

Theorempw2f1olem 7171* Lemma for pw2f1o 7172. (Contributed by Mario Carneiro, 6-Oct-2014.)

Theorempw2f1o 7172* The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)

Theorempw2eng 7173 The power set of a set is equinumerous to set exponentiation with a base of ordinal . (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)

Theorempw2en 7174 The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.)

Theoremfopwdom 7175 Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)

2.4.32  Schroeder-Bernstein Theorem

Theoremsbthlem1 7176* Lemma for sbth 7186. (Contributed by NM, 22-Mar-1998.)

Theoremsbthlem2 7177* Lemma for sbth 7186. (Contributed by NM, 22-Mar-1998.)

Theoremsbthlem3 7178* Lemma for sbth 7186. (Contributed by NM, 22-Mar-1998.)

Theoremsbthlem4 7179* Lemma for sbth 7186. (Contributed by NM, 27-Mar-1998.)

Theoremsbthlem5 7180* Lemma for sbth 7186. (Contributed by NM, 22-Mar-1998.)

Theoremsbthlem6 7181* Lemma for sbth 7186. (Contributed by NM, 27-Mar-1998.)

Theoremsbthlem7 7182* Lemma for sbth 7186. (Contributed by NM, 27-Mar-1998.)

Theoremsbthlem8 7183* Lemma for sbth 7186. (Contributed by NM, 27-Mar-1998.)

Theoremsbthlem9 7184* Lemma for sbth 7186. (Contributed by NM, 28-Mar-1998.)

Theoremsbthlem10 7185* Lemma for sbth 7186. (Contributed by NM, 28-Mar-1998.)

Theoremsbth 7186 Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set is smaller (has lower cardinality) than and vice-versa, then and are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7176 through sbthlem10 7185; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7185. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. (Contributed by NM, 8-Jun-1998.)

Theoremsbthb 7187 Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)

Theoremsbthcl 7188 Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)

Theoremdfsdom2 7189 Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)

Theorembrsdom2 7190 Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.)

Theoremsdomnsym 7191 Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)

Theoremdomnsym 7192 Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.)

Theorem0domg 7193 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremdom0 7194 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)

Theorem0sdomg 7195 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.)

Theorem0dom 7196 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theorem0sdom 7197 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)

Theoremsdom0 7198 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.)

Theoremsdomdomtr 7199 Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Theoremsdomentr 7200 Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)

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