HomeHome Intuitionistic Logic Explorer
Theorem List (p. 65 of 113)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 6401-6500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1oen2g 6401 The domain and range of a one-to-one, onto function are equinumerous. This variation of f1oeng 6403 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 10-Sep-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1dom2g 6402 The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6404 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
((𝐴𝑉𝐵𝑊𝐹:𝐴1-1𝐵) → 𝐴𝐵)
 
Theoremf1oeng 6403 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 6404 The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
(𝐵𝐶 → (𝐹:𝐴1-1𝐵𝐴𝐵))
 
Theoremf1oen 6405 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 6406 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 6407* 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 6408 Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.)
≈ ⊆ ≼
 
Theoremendom 6409 Equinumerosity implies dominance. Theorem 15 of [Suppes] p. 94. (Contributed by NM, 28-May-1998.)
(𝐴𝐵𝐴𝐵)
 
Theoremenrefg 6410 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉𝐴𝐴)
 
Theoremenref 6411 Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
𝐴 ∈ V       𝐴𝐴
 
Theoremeqeng 6412 Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)
(𝐴𝑉 → (𝐴 = 𝐵𝐴𝐵))
 
Theoremdomrefg 6413 Dominance is reflexive. (Contributed by NM, 18-Jun-1998.)
(𝐴𝑉𝐴𝐴)
 
Theoremen2d 6414* 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 6415* 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 6416* 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 6417* 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 6418* 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 6419* 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 6420* 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 6421* 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 6422* 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 6423 Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
I ⊆ ≈
 
Theoremssdomg 6424 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 6425 Equinumerosity is an equivalence relation. (Contributed by NM, 19-Mar-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)
≈ Er V
 
Theoremensymb 6426 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵𝐵𝐴)
 
Theoremensym 6427 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝐵𝐵𝐴)
 
Theoremensymi 6428 Symmetry of equinumerosity. Theorem 2 of [Suppes] p. 92. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵       𝐵𝐴
 
Theoremensymd 6429 Symmetry of equinumerosity. Deduction form of ensym 6427. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑𝐵𝐴)
 
Theorementr 6430 Transitivity of equinumerosity. Theorem 3 of [Suppes] p. 92. (Contributed by NM, 9-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomtr 6431 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 6432 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐵𝐶       𝐴𝐶
 
Theorementr2i 6433 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐵𝐶       𝐶𝐴
 
Theorementr3i 6434 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐴𝐶       𝐵𝐶
 
Theorementr4i 6435 A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.)
𝐴𝐵    &   𝐶𝐵       𝐴𝐶
 
Theoremendomtr 6436 Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomentr 6437 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremf1imaeng 6438 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 6439 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6440 does not need ax-setind 4315.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
(((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)
 
Theoremf1imaen 6440 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 6441 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)
 
Theoremensn1 6442 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
𝐴 ∈ V       {𝐴} ≈ 1𝑜
 
Theoremensn1g 6443 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → {𝐴} ≈ 1𝑜)
 
Theoremenpr1g 6444 {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
(𝐴𝑉 → {𝐴, 𝐴} ≈ 1𝑜)
 
Theoremen1 6445* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
(𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
 
Theoremen1bg 6446 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
(𝐴𝑉 → (𝐴 ≈ 1𝑜𝐴 = { 𝐴}))
 
Theoremreuen1 6447* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝐴 𝜑 ↔ {𝑥𝐴𝜑} ≈ 1𝑜)
 
Theoremeuen1 6448 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1𝑜)
 
Theoremeuen1b 6449* Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥𝐴)
 
Theoremen1uniel 6450 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(𝑆 ≈ 1𝑜 𝑆𝑆)
 
Theorem2dom 6451* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
(2𝑜𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)
 
Theoremfundmen 6452 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 6453 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 6454 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
((Rel 𝐴𝐴𝑉) → 𝐴𝐴)
 
Theoremcnvct 6455 If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → 𝐴 ≼ ω)
 
Theoremfndmeng 6456 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)
 
Theoremmapsnen 6457 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 6458 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 6459 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})
 
Theoremsnfig 6460 A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.)
(𝐴𝑉 → {𝐴} ∈ Fin)
 
Theoremfiprc 6461 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
Fin ∉ V
 
Theoremunen 6462 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))
 
Theoremssct 6463 Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)
 
Theorem1domsn 6464 A singleton (whether of a set or a proper class) is dominated by one. (Contributed by Jim Kingdon, 1-Mar-2022.)
{𝐴} ≼ 1𝑜
 
Theoremenm 6465* A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.)
((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴) → ∃𝑦 𝑦𝐵)
 
Theoremxpsnen 6466 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 6467 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 6468 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 × 1𝑜) ≈ 𝐴)
 
Theoremendisj 6469* 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 6470* 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 6471* Composition with the bijection of xpcomf1o 6470 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 6472 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 6473 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 6474 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 6475 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 6476 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 6477 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 6478 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 6479* 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 6480 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 6481 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 6482 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉 → ∅ ≼ 𝐴)
 
Theoremdom0 6483 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)
 
Theorem0dom 6484 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ∅ ≼ 𝐴
 
Theoremenen1 6485 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremenen2 6486 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremdomen1 6487 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremdomen2 6488 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
2.6.27  Equinumerosity (cont.)
 
Theoremxpf1o 6489* Construct a bijection on a Cartesian product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)
(𝜑 → (𝑥𝐴𝑋):𝐴1-1-onto𝐵)    &   (𝜑 → (𝑦𝐶𝑌):𝐶1-1-onto𝐷)       (𝜑 → (𝑥𝐴, 𝑦𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))
 
Theoremxpen 6490 Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
 
Theoremmapen 6491 Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐶𝐷) → (𝐴𝑚 𝐶) ≈ (𝐵𝑚 𝐷))
 
Theoremmapdom1g 6492 Order-preserving property of set exponentiation. (Contributed by Jim Kingdon, 15-Jul-2022.)
((𝐴𝐵𝐶𝑉) → (𝐴𝑚 𝐶) ≼ (𝐵𝑚 𝐶))
 
Theoremmapxpen 6493 Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝐴𝑚 𝐵) ↑𝑚 𝐶) ≈ (𝐴𝑚 (𝐵 × 𝐶)))
 
Theoremxpmapenlem 6494* Lemma for xpmapen 6495. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V    &   𝐷 = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))    &   𝑅 = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))    &   𝑆 = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)       ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))
 
Theoremxpmapen 6495 Equinumerosity law for set exponentiation of a Cartesian product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))
 
Theoremssenen 6496* Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
(𝐴𝐵 → {𝑥 ∣ (𝑥𝐴𝑥𝐶)} ≈ {𝑥 ∣ (𝑥𝐵𝑥𝐶)})
 
2.6.28  Pigeonhole Principle
 
Theoremphplem1 6497 Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))
 
Theoremphplem2 6498 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
 
Theoremphplem3 6499 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6501. (Contributed by NM, 26-May-1998.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
 
Theoremphplem4 6500 Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11237
  Copyright terms: Public domain < Previous  Next >