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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvixp 8201* | Projection of a factor of an indexed Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) | ||
Theorem | ixpfn 8202* | A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-May-2014.) |
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) | ||
Theorem | elixp 8203* | Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
Theorem | elixpconst 8204* | Membership in an infinite Cartesian product of a constant 𝐵. (Contributed by NM, 12-Apr-2008.) |
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵) | ||
Theorem | ixpconstg 8205* | Infinite Cartesian product of a constant 𝐵. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑𝑚 𝐴)) | ||
Theorem | ixpconst 8206* | Infinite Cartesian product of a constant 𝐵. (Contributed by NM, 28-Sep-2006.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑𝑚 𝐴) | ||
Theorem | ixpeq1 8207* | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | ixpeq1d 8208* | Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | ss2ixp 8209 | Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2 8210 | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2dva 8211* | Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2dv 8212* | Equality theorem for infinite Cartesian product. (Contributed by Mario Carneiro, 11-Jun-2016.) |
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | cbvixp 8213* | Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 20-Jun-2011.) |
⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 | ||
Theorem | cbvixpv 8214* | Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 | ||
Theorem | nfixp 8215 | Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfixp1 8216 | The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | ixpprc 8217* | A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain 𝐴, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.) |
⊢ (¬ 𝐴 ∈ V → X𝑥 ∈ 𝐴 𝐵 = ∅) | ||
Theorem | ixpf 8218* | A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.) |
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | uniixp 8219* | The union of an infinite Cartesian product is included in a Cartesian product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ ∪ X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) | ||
Theorem | ixpexg 8220* | The existence of an infinite Cartesian product. 𝑥 is normally a free-variable parameter in 𝐵. Remark in Enderton p. 54. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro, 25-Jan-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ∈ V) | ||
Theorem | ixpin 8221* | The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) |
⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpiin 8222* | The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) |
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpint 8223* | The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) |
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) | ||
Theorem | ixp0x 8224 | An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.) |
⊢ X𝑥 ∈ ∅ 𝐴 = {∅} | ||
Theorem | ixpssmap2g 8225* | An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8226 avoids ax-rep 5008. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) | ||
Theorem | ixpssmapg 8226* | An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) | ||
Theorem | 0elixp 8227 | Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.) |
⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴 | ||
Theorem | ixpn0 8228 | The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9642. (Contributed by Mario Carneiro, 22-Jun-2016.) |
⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) | ||
Theorem | ixp0 8229 | The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9642. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.) |
⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅) | ||
Theorem | ixpssmap 8230* | An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.) |
⊢ 𝐵 ∈ V ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) | ||
Theorem | resixp 8231* | Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.) |
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | undifixp 8232* | Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.) |
⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) ∈ X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | mptelixpg 8233* | Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.) |
⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾)) | ||
Theorem | resixpfo 8234* | Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝐹 = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↦ (𝑓 ↾ 𝐵)) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | elixpsn 8235* | Membership in a class of singleton functions. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {〈𝐴, 𝑦〉})) | ||
Theorem | ixpsnf1o 8236* | A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥})) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴) | ||
Theorem | mapsnf1o 8237* | A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥})) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑𝑚 {𝐼})) | ||
Theorem | boxriin 8238* | A rectangular subset of a rectangular set can be recovered as the relative intersection of single-axis restrictions. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → X𝑥 ∈ 𝐼 𝐴 = (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))) | ||
Theorem | boxcutc 8239* | The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑘 ∈ 𝐴 𝐵 ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, 𝐶, 𝐵)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, (𝐵 ∖ 𝐶), 𝐵)) | ||
Syntax | cen 8240 | Extend class definition to include the equinumerosity relation ("approximately equals" symbol) |
class ≈ | ||
Syntax | cdom 8241 | Extend class definition to include the dominance relation (curly "less than or equal to") |
class ≼ | ||
Syntax | csdm 8242 | Extend class definition to include the strict dominance relation (curly less-than) |
class ≺ | ||
Syntax | cfn 8243 | Extend class definition to include the class of all finite sets. |
class Fin | ||
Definition | df-en 8244* | 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 8252. (Contributed by NM, 28-Mar-1998.) |
⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | ||
Definition | df-dom 8245* | Define the dominance relation. For an alternate definition see dfdom2 8269. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 8255 and domen 8256. (Contributed by NM, 28-Mar-1998.) |
⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | ||
Definition | df-sdom 8246 | Define the strict dominance relation. Alternate possible definitions are derived as brsdom 8266 and brsdom2 8374. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
⊢ ≺ = ( ≼ ∖ ≈ ) | ||
Definition | df-fin 8247* | 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 8837. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (theorem isfinite 8848.) (Contributed by NM, 22-Aug-2008.) |
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} | ||
Theorem | relen 8248 | Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≈ | ||
Theorem | reldom 8249 | Dominance is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≼ | ||
Theorem | relsdom 8250 | Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.) |
⊢ Rel ≺ | ||
Theorem | encv 8251 | If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | bren 8252* | Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) |
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) | ||
Theorem | brdomg 8253* | Dominance relation. (Contributed by NM, 15-Jun-1998.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | ||
Theorem | brdomi 8254* | Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | brdom 8255* | Dominance relation. (Contributed by NM, 15-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | domen 8256* | Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) | ||
Theorem | domeng 8257* | 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.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) | ||
Theorem | ctex 8258 | A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) (Proof shortened by Jim Kingdon, 13-Mar-2023.) |
⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | ||
Theorem | f1oen3g 8259 | The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8262 does not require the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1oen2g 8260 | The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8262 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1dom2g 8261 | The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8263 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵) | ||
Theorem | f1oeng 8262 | The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵) | ||
Theorem | f1domg 8263 | The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.) |
⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵)) | ||
Theorem | f1oen 8264 | The domain and range of a one-to-one, onto function are equinumerous. (Contributed by NM, 19-Jun-1998.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐴 ≈ 𝐵) | ||
Theorem | f1dom 8265 | The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 19-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | brsdom 8266 | Strict dominance relation, meaning "𝐵 is strictly greater in size than 𝐴". Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | ||
Theorem | isfi 8267* | 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 𝐴 ≈ 𝑥) | ||
Theorem | enssdom 8268 | Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) |
⊢ ≈ ⊆ ≼ | ||
Theorem | dfdom2 8269 | Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.) |
⊢ ≼ = ( ≺ ∪ ≈ ) | ||
Theorem | endom 8270 | Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.) |
⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomdom 8271 | Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomnen 8272 | Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | brdom2 8273 | Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.) |
⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | ||
Theorem | bren2 8274 | Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) | ||
Theorem | enrefg 8275 | Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ 𝐴) | ||
Theorem | enref 8276 | Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ≈ 𝐴 | ||
Theorem | eqeng 8277 | Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵)) | ||
Theorem | domrefg 8278 | Dominance is reflexive. (Contributed by NM, 18-Jun-1998.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴) | ||
Theorem | en2d 8279* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ V)) & ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ V)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷))) ⇒ ⊢ (𝜑 → 𝐴 ≈ 𝐵) | ||
Theorem | en3d 8280* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 12-May-2014.) |
⊢ (𝜑 → 𝐴 ∈ V) & ⊢ (𝜑 → 𝐵 ∈ V) & ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))) ⇒ ⊢ (𝜑 → 𝐴 ≈ 𝐵) | ||
Theorem | en2i 8281* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 4-Jan-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ V) & ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ V) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = 𝐷)) ⇒ ⊢ 𝐴 ≈ 𝐵 | ||
Theorem | en3i 8282* | Equinumerosity inference from an implicit one-to-one onto function. (Contributed by NM, 19-Jul-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) & ⊢ (𝑦 ∈ 𝐵 → 𝐷 ∈ 𝐴) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶)) ⇒ ⊢ 𝐴 ≈ 𝐵 | ||
Theorem | dom2lem 8283* | 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 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴–1-1→𝐵) | ||
Theorem | dom2d 8284* | 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 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) ⇒ ⊢ (𝜑 → (𝐵 ∈ 𝑅 → 𝐴 ≼ 𝐵)) | ||
Theorem | dom3d 8285* | 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 → (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵)) & ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦))) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝐴 ≼ 𝐵) | ||
Theorem | dom2 8286* | 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.) |
⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)) ⇒ ⊢ (𝐵 ∈ 𝑉 → 𝐴 ≼ 𝐵) | ||
Theorem | dom3 8287* | 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.) |
⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐶 = 𝐷 ↔ 𝑥 = 𝑦)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ 𝐵) | ||
Theorem | idssen 8288 | Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ I ⊆ ≈ | ||
Theorem | ssdomg 8289 | A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴 ⊆ 𝐵 → 𝐴 ≼ 𝐵)) | ||
Theorem | ener 8290 | Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof shortened by AV, 1-May-2021.) |
⊢ ≈ Er V | ||
Theorem | ensymb 8291 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) | ||
Theorem | ensym 8292 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | ||
Theorem | ensymi 8293 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 ⇒ ⊢ 𝐵 ≈ 𝐴 | ||
Theorem | ensymd 8294 | Symmetry of equinumerosity. Deduction form of ensym 8292. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≈ 𝐴) | ||
Theorem | entr 8295 | Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | ||
Theorem | domtr 8296 | Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | entri 8297 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐴 ≈ 𝐶 | ||
Theorem | entr2i 8298 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐶 ≈ 𝐴 | ||
Theorem | entr3i 8299 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐴 ≈ 𝐶 ⇒ ⊢ 𝐵 ≈ 𝐶 | ||
Theorem | entr4i 8300 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐶 ≈ 𝐵 ⇒ ⊢ 𝐴 ≈ 𝐶 |
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