Home | Metamath
Proof Explorer Theorem List (p. 86 of 450) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28698) |
Hilbert Space Explorer
(28699-30221) |
Users' Mathboxes
(30222-44913) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | undifixp 8501* | 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 8502* | 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 8503* | 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 8504* | 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 8505* | 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 8506* | 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 8507* | 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 8508* | 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 8509 | Extend class definition to include the equinumerosity relation ("approximately equals" symbol) |
class ≈ | ||
Syntax | cdom 8510 | Extend class definition to include the dominance relation (curly "less than or equal to") |
class ≼ | ||
Syntax | csdm 8511 | Extend class definition to include the strict dominance relation (curly less-than) |
class ≺ | ||
Syntax | cfn 8512 | Extend class definition to include the class of all finite sets. |
class Fin | ||
Definition | df-en 8513* | 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 8521. (Contributed by NM, 28-Mar-1998.) |
⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} | ||
Definition | df-dom 8514* | Define the dominance relation. For an alternate definition see dfdom2 8538. Compare Definition of [Enderton] p. 145. Typical textbook definitions are derived as brdom 8524 and domen 8525. (Contributed by NM, 28-Mar-1998.) |
⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} | ||
Definition | df-sdom 8515 | Define the strict dominance relation. Alternate possible definitions are derived as brsdom 8535 and brsdom2 8644. Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.) |
⊢ ≺ = ( ≼ ∖ ≈ ) | ||
Definition | df-fin 8516* | 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 9107. If we accept Infinity, we can also express 𝐴 ∈ Fin by 𝐴 ≺ ω (theorem isfinite 9118.) (Contributed by NM, 22-Aug-2008.) |
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} | ||
Theorem | relen 8517 | Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≈ | ||
Theorem | reldom 8518 | Dominance is a relation. (Contributed by NM, 28-Mar-1998.) |
⊢ Rel ≼ | ||
Theorem | relsdom 8519 | Strict dominance is a relation. (Contributed by NM, 31-Mar-1998.) |
⊢ Rel ≺ | ||
Theorem | encv 8520 | If two classes are equinumerous, both classes are sets. (Contributed by AV, 21-Mar-2019.) |
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | ||
Theorem | bren 8521* | Equinumerosity relation. (Contributed by NM, 15-Jun-1998.) |
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) | ||
Theorem | brdomg 8522* | Dominance relation. (Contributed by NM, 15-Jun-1998.) |
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) | ||
Theorem | brdomi 8523* | Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | brdom 8524* | Dominance relation. (Contributed by NM, 15-Jun-1998.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) | ||
Theorem | domen 8525* | 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 8526* | 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 8527 | 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 8528 | The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8531 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 8529 | The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 8531 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 8530 | The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 8532 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 8531 | 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 8532 | 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 8533 | 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 8534 | 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 8535 | 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 8536* | 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 8537 | Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) |
⊢ ≈ ⊆ ≼ | ||
Theorem | dfdom2 8538 | Alternate definition of dominance. (Contributed by NM, 17-Jun-1998.) |
⊢ ≼ = ( ≺ ∪ ≈ ) | ||
Theorem | endom 8539 | Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.) |
⊢ (𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomdom 8540 | Strict dominance implies dominance. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵) | ||
Theorem | sdomnen 8541 | Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.) |
⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | brdom2 8542 | 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 8543 | Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ≈ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≺ 𝐵)) | ||
Theorem | enrefg 8544 | 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 8545 | Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ 𝐴 ≈ 𝐴 | ||
Theorem | eqeng 8546 | Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵)) | ||
Theorem | domrefg 8547 | Dominance is reflexive. (Contributed by NM, 18-Jun-1998.) |
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝐴) | ||
Theorem | en2d 8548* | 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 8549* | 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 8550* | 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 8551* | 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 8552* | 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 8553* | 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 8554* | 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 8555* | 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 8556* | 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 8557 | Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ I ⊆ ≈ | ||
Theorem | ssdomg 8558 | 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 8559 | 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 8560 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴) | ||
Theorem | ensym 8561 | 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 8562 | Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 ⇒ ⊢ 𝐵 ≈ 𝐴 | ||
Theorem | ensymd 8563 | Symmetry of equinumerosity. Deduction form of ensym 8561. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ≈ 𝐵) ⇒ ⊢ (𝜑 → 𝐵 ≈ 𝐴) | ||
Theorem | entr 8564 | Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶) | ||
Theorem | domtr 8565 | 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 8566 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐴 ≈ 𝐶 | ||
Theorem | entr2i 8567 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐵 ≈ 𝐶 ⇒ ⊢ 𝐶 ≈ 𝐴 | ||
Theorem | entr3i 8568 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐴 ≈ 𝐶 ⇒ ⊢ 𝐵 ≈ 𝐶 | ||
Theorem | entr4i 8569 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
⊢ 𝐴 ≈ 𝐵 & ⊢ 𝐶 ≈ 𝐵 ⇒ ⊢ 𝐴 ≈ 𝐶 | ||
Theorem | endomtr 8570 | Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | domentr 8571 | Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.) |
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) | ||
Theorem | f1imaeng 8572 | 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:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐹 “ 𝐶) ≈ 𝐶) | ||
Theorem | f1imaen2g 8573 | A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 8574 does not need ax-reg 9059.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶) | ||
Theorem | f1imaen 8574 | 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:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 “ 𝐶) ≈ 𝐶) | ||
Theorem | en0 8575 | The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) |
⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) | ||
Theorem | ensn1 8576 | A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) |
⊢ 𝐴 ∈ V ⇒ ⊢ {𝐴} ≈ 1o | ||
Theorem | ensn1g 8577 | A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | ||
Theorem | enpr1g 8578 | {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.) |
⊢ (𝐴 ∈ 𝑉 → {𝐴, 𝐴} ≈ 1o) | ||
Theorem | en1 8579* | A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) |
⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | en1b 8580 | A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) |
⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) | ||
Theorem | reuen1 8581* | Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ {𝑥 ∈ 𝐴 ∣ 𝜑} ≈ 1o) | ||
Theorem | euen1 8582 | Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) | ||
Theorem | euen1b 8583* | Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.) |
⊢ (𝐴 ≈ 1o ↔ ∃!𝑥 𝑥 ∈ 𝐴) | ||
Theorem | en1uniel 8584 | A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) | ||
Theorem | 2dom 8585* | A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.) |
⊢ (2o ≼ 𝐴 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦) | ||
Theorem | fundmen 8586 | 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 𝐹 ≈ 𝐹) | ||
Theorem | fundmeng 8587 | A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.) |
⊢ ((𝐹 ∈ 𝑉 ∧ Fun 𝐹) → dom 𝐹 ≈ 𝐹) | ||
Theorem | cnven 8588 | A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.) |
⊢ ((Rel 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐴 ≈ ◡𝐴) | ||
Theorem | cnvct 8589 | If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → ◡𝐴 ≼ ω) | ||
Theorem | fndmeng 8590 | A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐶) → 𝐴 ≈ 𝐹) | ||
Theorem | mapsnend 8591 | 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.) (Revised by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 ↑m {𝐵}) ≈ 𝐴) | ||
Theorem | mapsnen 8592 | 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.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ↑m {𝐵}) ≈ 𝐴 | ||
Theorem | snmapen 8593 | Set exponentiation: a singleton to any set is equinumerous to that singleton. (Contributed by NM, 17-Dec-2003.) (Revised by AV, 17-Jul-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ {𝐴}) | ||
Theorem | snmapen1 8594 | Set exponentiation: a singleton to any set is equinumerous to ordinal 1. (Proposed by BJ, 17-Jul-2022.) (Contributed by AV, 17-Jul-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} ↑m 𝐵) ≈ 1o) | ||
Theorem | map1 8595 | 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.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ (𝐴 ∈ 𝑉 → (1o ↑m 𝐴) ≈ 1o) | ||
Theorem | en2sn 8596 | Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) | ||
Theorem | snfi 8597 | A singleton is finite. (Contributed by NM, 4-Nov-2002.) |
⊢ {𝐴} ∈ Fin | ||
Theorem | fiprc 8598 | The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.) |
⊢ Fin ∉ V | ||
Theorem | unen 8599 | 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.) |
⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)) | ||
Theorem | enpr2d 8600 | A pair with distinct elements is equinumerous to ordinal two. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → ¬ 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → {𝐴, 𝐵} ≈ 2o) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |