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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcdom 6601 Extend class definition to include the dominance relation (curly less-than-or-equal)
class
 
Syntaxcfn 6602 Extend class definition to include the class of all finite sets.
class Fin
 
Definitiondf-en 6603* Define the equinumerosity relation. Definition of [Enderton] p. 129. We define to be a binary relation rather than a connective, so its arguments must be sets to be meaningful. This is acceptable because we do not consider equinumerosity for proper classes. We derive the usual definition as bren 6609. (Contributed by NM, 28-Mar-1998.)
≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1-onto𝑦}
 
Definitiondf-dom 6604* Define the dominance relation. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 6612 and domen 6613. (Contributed by NM, 28-Mar-1998.)
≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥1-1𝑦}
 
Definitiondf-fin 6605* Define the (proper) class of all finite sets. Similar to Definition 10.29 of [TakeutiZaring] p. 91, whose "Fin(a)" corresponds to our "𝑎 ∈ Fin". This definition is meaningful whether or not we accept the Axiom of Infinity ax-inf2 13101. (Contributed by NM, 22-Aug-2008.)
Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥𝑦}
 
Theoremrelen 6606 Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
Rel ≈
 
Theoremreldom 6607 Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
Rel ≼
 
Theoremencv 6608 If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.)
(𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
 
Theorembren 6609* Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
(𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1-onto𝐵)
 
Theorembrdomg 6610* Dominance relation. (Contributed by NM, 15-Jun-1998.)
(𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
 
Theorembrdomi 6611* Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
 
Theorembrdom 6612* Dominance relation. (Contributed by NM, 15-Jun-1998.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)
 
Theoremdomen 6613* Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵))
 
Theoremdomeng 6614* Dominance in terms of equinumerosity, with the sethood requirement expressed as an antecedent. Example 1 of [Enderton] p. 146. (Contributed by NM, 24-Apr-2004.)
(𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑥(𝐴𝑥𝑥𝐵)))
 
Theoremctex 6615 A class dominated by ω is a set. See also ctfoex 6971 which says that a countable class is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.)
(𝐴 ≼ ω → 𝐴 ∈ V)
 
Theoremf1oen3g 6616 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6619 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.)
((𝐹𝑉𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1oen2g 6617 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6619 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1dom2g 6618 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6620 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
 
Theoremf1oeng 6619 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
((𝐴𝐶𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1domg 6620 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
(𝐵𝐶 → (𝐹:𝐴1-1𝐵𝐴𝐵))
 
Theoremf1oen 6621 The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.)
𝐴 ∈ V       (𝐹:𝐴1-1-onto𝐵𝐴𝐵)
 
Theoremf1dom 6622 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.)
𝐵 ∈ V       (𝐹:𝐴1-1𝐵𝐴𝐵)
 
Theoremisfi 6623* Express "𝐴 is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
(𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
 
Theoremenssdom 6624 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
≈ ⊆ ≼
 
Theoremendom 6625 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
(𝐴𝐵𝐴𝐵)
 
Theoremenrefg 6626 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉𝐴𝐴)
 
Theoremenref 6627 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
𝐴 ∈ V       𝐴𝐴
 
Theoremeqeng 6628 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
(𝐴𝑉 → (𝐴 = 𝐵𝐴𝐵))
 
Theoremdomrefg 6629 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
(𝐴𝑉𝐴𝐴)
 
Theoremen2d 6630* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑥𝐴𝐶 ∈ V))    &   (𝜑 → (𝑦𝐵𝐷 ∈ V))    &   (𝜑 → ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷)))       (𝜑𝐴𝐵)
 
Theoremen3d 6631* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.)
(𝜑𝐴 ∈ V)    &   (𝜑𝐵 ∈ V)    &   (𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → (𝑦𝐵𝐷𝐴))    &   (𝜑 → ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶)))       (𝜑𝐴𝐵)
 
Theoremen2i 6632* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐴𝐶 ∈ V)    &   (𝑦𝐵𝐷 ∈ V)    &   ((𝑥𝐴𝑦 = 𝐶) ↔ (𝑦𝐵𝑥 = 𝐷))       𝐴𝐵
 
Theoremen3i 6633* Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝑥𝐴𝐶𝐵)    &   (𝑦𝐵𝐷𝐴)    &   ((𝑥𝐴𝑦𝐵) → (𝑥 = 𝐷𝑦 = 𝐶))       𝐴𝐵
 
Theoremdom2lem 6634* 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.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))       (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
 
Theoremdom2d 6635* 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.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))       (𝜑 → (𝐵𝑅𝐴𝐵))
 
Theoremdom3d 6636* 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.)
(𝜑 → (𝑥𝐴𝐶𝐵))    &   (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑𝐴𝐵)
 
Theoremdom2 6637* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by NM, 26-Oct-2003.)
(𝑥𝐴𝐶𝐵)    &   ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))       (𝐵𝑉𝐴𝐵)
 
Theoremdom3 6638* A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
(𝑥𝐴𝐶𝐵)    &   ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))       ((𝐴𝑉𝐵𝑊) → 𝐴𝐵)
 
Theoremidssen 6639 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
I ⊆ ≈
 
Theoremssdomg 6640 A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
(𝐵𝑉 → (𝐴𝐵𝐴𝐵))
 
Theoremener 6641 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
≈ Er V
 
Theoremensymb 6642 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵𝐵𝐴)
 
Theoremensym 6643 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵𝐵𝐴)
 
Theoremensymi 6644 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵       𝐵𝐴
 
Theoremensymd 6645 Symmetry of equinumerosity. Deduction form of ensym 6643. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)
 
Theorementr 6646 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomtr 6647 Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theorementri 6648 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶
 
Theorementr2i 6649 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐵𝐶       𝐶𝐴
 
Theorementr3i 6650 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐴𝐶       𝐵𝐶
 
Theorementr4i 6651 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐶𝐵       𝐴𝐶
 
Theoremendomtr 6652 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomentr 6653 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremf1imaeng 6654 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.)
((𝐹:𝐴1-1𝐵𝐶𝐴𝐶𝑉) → (𝐹𝐶) ≈ 𝐶)
 
Theoremf1imaen2g 6655 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6656 does not need ax-setind 4422.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
(((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)
 
Theoremf1imaen 6656 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.)
𝐶 ∈ V       ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶) ≈ 𝐶)
 
Theoremen0 6657 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)
 
Theoremensn1 6658 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
𝐴 ∈ V       {𝐴} ≈ 1o
 
Theoremensn1g 6659 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → {𝐴} ≈ 1o)
 
Theoremenpr1g 6660 {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
(𝐴𝑉 → {𝐴, 𝐴} ≈ 1o)
 
Theoremen1 6661* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
(𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremen1bg 6662 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
(𝐴𝑉 → (𝐴 ≈ 1o𝐴 = { 𝐴}))
 
Theoremreuen1 6663* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝐴 𝜑 ↔ {𝑥𝐴𝜑} ≈ 1o)
 
Theoremeuen1 6664 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1o)
 
Theoremeuen1b 6665* Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴 ≈ 1o ↔ ∃!𝑥 𝑥𝐴)
 
Theoremen1uniel 6666 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(𝑆 ≈ 1o 𝑆𝑆)
 
Theorem2dom 6667* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
(2o𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
 
Theoremfundmen 6668 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.)
𝐹 ∈ V       (Fun 𝐹 → dom 𝐹𝐹)
 
Theoremfundmeng 6669 A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)
((𝐹𝑉 ∧ Fun 𝐹) → dom 𝐹𝐹)
 
Theoremcnven 6670 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
((Rel 𝐴𝐴𝑉) → 𝐴𝐴)
 
Theoremcnvct 6671 If a set is dominated by ω, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → 𝐴 ≼ ω)
 
Theoremfndmeng 6672 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)
 
Theoremmapsnen 6673 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.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝑚 {𝐵}) ≈ 𝐴
 
Theoremmap1 6674 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.)
(𝐴𝑉 → (1o𝑚 𝐴) ≈ 1o)
 
Theoremen2sn 6675 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})
 
Theoremsnfig 6676 A singleton is finite. For the proper class case, see snprc 3558. (Contributed by Jim Kingdon, 13-Apr-2020.)
(𝐴𝑉 → {𝐴} ∈ Fin)
 
Theoremfiprc 6677 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
Fin ∉ V
 
Theoremunen 6678 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.)
(((𝐴𝐵𝐶𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐴𝐶) ≈ (𝐵𝐷))
 
Theoremenpr2d 6679 A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐷)    &   (𝜑 → ¬ 𝐴 = 𝐵)       (𝜑 → {𝐴, 𝐵} ≈ 2o)
 
Theoremssct 6680 A subset of a set dominated by ω is dominated by ω. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)
 
Theorem1domsn 6681 A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
{𝐴} ≼ 1o
 
Theoremenm 6682* A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → ∃𝑦 𝑦𝐵)
 
Theoremxpsnen 6683 A set is equinumerous to its Cartesian 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.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 × {𝐵}) ≈ 𝐴
 
Theoremxpsneng 6684 A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴 × {𝐵}) ≈ 𝐴)
 
Theoremxp1en 6685 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 × 1o) ≈ 𝐴)
 
Theoremendisj 6686* Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑥𝑦) = ∅)
 
Theoremxpcomf1o 6687* The canonical bijection from (𝐴 × 𝐵) to (𝐵 × 𝐴). (Contributed by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})       𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
 
Theoremxpcomco 6688* Composition with the bijection of xpcomf1o 6687 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ {𝑥})    &   𝐺 = (𝑦𝐵, 𝑧𝐴𝐶)       (𝐺𝐹) = (𝑧𝐴, 𝑦𝐵𝐶)
 
Theoremxpcomen 6689 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴 × 𝐵) ≈ (𝐵 × 𝐴)
 
Theoremxpcomeng 6690 Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.)
((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ≈ (𝐵 × 𝐴))
 
Theoremxpsnen2g 6691 A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.)
((𝐴𝑉𝐵𝑊) → ({𝐴} × 𝐵) ≈ 𝐵)
 
Theoremxpassen 6692 Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((𝐴 × 𝐵) × 𝐶) ≈ (𝐴 × (𝐵 × 𝐶))
 
Theoremxpdom2 6693 Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
𝐶 ∈ V       (𝐴𝐵 → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
 
Theoremxpdom2g 6694 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.)
((𝐶𝑉𝐴𝐵) → (𝐶 × 𝐴) ≼ (𝐶 × 𝐵))
 
Theoremxpdom1g 6695 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐶𝑉𝐴𝐵) → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
 
Theoremxpdom3m 6696* A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.)
((𝐴𝑉𝐵𝑊 ∧ ∃𝑥 𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
 
Theoremxpdom1 6697 Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)
𝐶 ∈ V       (𝐴𝐵 → (𝐴 × 𝐶) ≼ (𝐵 × 𝐶))
 
Theoremfopwdom 6698 Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹 ∈ V ∧ 𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)
 
Theorem0domg 6699 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉 → ∅ ≼ 𝐴)
 
Theoremdom0 6700 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)
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