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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | elixp 7801* | 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 7802* | 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 7803* | 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 7804* | 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 7805* | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | ixpeq1d 7806* | 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 7807 | 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 7808 | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2dva 7809* | 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 7810* | 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 7811* | 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 7812* | 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 7813 | Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfixp1 7814 | 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 7815* | 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 7816* | 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 7817* | 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 7818* | 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 7819* | 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 7820* | 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 7821* | 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 7822 | An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.) |
⊢ X𝑥 ∈ ∅ 𝐴 = {∅} | ||
Theorem | ixpssmap2g 7823* | An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7824 avoids ax-rep 4699. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) | ||
Theorem | ixpssmapg 7824* | 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 7825 | Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.) |
⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴 | ||
Theorem | ixpn0 7826 | 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 9188. (Contributed by Mario Carneiro, 22-Jun-2016.) |
⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) | ||
Theorem | ixp0 7827 | 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 9188. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.) |
⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅) | ||
Theorem | ixpssmap 7828* | 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 7829* | 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 7830* | 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 7831* | 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 7832* | 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 7833* | 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 7834* | 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 7835* | 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 7836* | 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 7837* | 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 7838 | Extend class definition to include the equinumerosity relation ("approximately equals" symbol) |
class ≈ | ||
Syntax | cdom 7839 | Extend class definition to include the dominance relation (curly less-than-or-equal) |
class ≼ | ||
Syntax | csdm 7840 | Extend class definition to include the strict dominance relation (curly less-than) |
class ≺ | ||
Syntax | cfn 7841 | Extend class definition to include the class of all finite sets. |
class Fin | ||
Definition | df-en 7842* | 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 7850. (Contributed by NM, 28-Mar-1998.) |
⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | ||
Definition | df-dom 7843* | Define the dominance relation. For an alternate definition see dfdom2 7867. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 7853 and domen 7854. (Contributed by NM, 28-Mar-1998.) |
⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | ||
Definition | df-sdom 7844 | Define the strict dominance relation. Alternate possible definitions are derived as brsdom 7864 and brsdom2 7969. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
⊢ ≺ = ( ≼ ∖ ≈ ) | ||
Definition | df-fin 7845* | 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 8421. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (theorem isfinite 8432.) (Contributed by NM, 22-Aug-2008.) |
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} | ||
Theorem | relen 7846 | Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≈ | ||
Theorem | reldom 7847 | Dominance is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≼ | ||
Theorem | relsdom 7848 | Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.) |
⊢ Rel ≺ | ||
Theorem | encv 7849 | If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | bren 7850* | Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) |
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) | ||
Theorem | brdomg 7851* | Dominance relation. (Contributed by NM, 15-Jun-1998.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | ||
Theorem | brdomi 7852* | Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | brdom 7853* | Dominance relation. (Contributed by NM, 15-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | domen 7854* | 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 7855* | 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 7856 | A countable set is a set. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → 𝐴 ∈ V) | ||
Theorem | f1oen3g 7857 | The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7860 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 7858 | The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 7860 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 7859 | The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 7861 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 7860 | 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 7861 | 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 7862 | 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 7863 | 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 7864 | 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 7865* | 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 7866 | Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) |
⊢ ≈ ⊆ ≼ | ||
Theorem | dfdom2 7867 | Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.) |
⊢ ≼ = ( ≺ ∪ ≈ ) | ||
Theorem | endom 7868 | Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.) |
⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomdom 7869 | Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomnen 7870 | Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | brdom2 7871 | 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 7872 | Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) | ||
Theorem | enrefg 7873 | 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 7874 | Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ≈ 𝐴 | ||
Theorem | eqeng 7875 | Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵)) | ||
Theorem | domrefg 7876 | Dominance is reflexive. (Contributed by NM, 18-Jun-1998.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴) | ||
Theorem | en2d 7877* | 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 7878* | 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 7879* | 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 7880* | 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 7881* | 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 7882* | 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 7883* | 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 7884* | 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 7885* | 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 7886 | Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ I ⊆ ≈ | ||
Theorem | ssdomg 7887 | 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 7888 | 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 | enerOLD 7889 | Obsolete proof of ener 7888 as of 1-May-2021. Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ≈ Er V | ||
Theorem | ensymb 7890 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) | ||
Theorem | ensym 7891 | 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 7892 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 ⇒ ⊢ 𝐵 ≈ 𝐴 | ||
Theorem | ensymd 7893 | Symmetry of equinumerosity. Deduction form of ensym 7891. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≈ 𝐴) | ||
Theorem | entr 7894 | Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | ||
Theorem | domtr 7895 | 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 7896 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐴 ≈ 𝐶 | ||
Theorem | entr2i 7897 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐶 ≈ 𝐴 | ||
Theorem | entr3i 7898 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐴 ≈ 𝐶 ⇒ ⊢ 𝐵 ≈ 𝐶 | ||
Theorem | entr4i 7899 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐶 ≈ 𝐵 ⇒ ⊢ 𝐴 ≈ 𝐶 | ||
Theorem | endomtr 7900 | Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) |
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