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Theorem List for Metamath Proof Explorer - 7901-8000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdomentr 7901 Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremf1imaeng 7902 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 7903 A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 7904 does not need ax-reg 8380.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
(((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)

Theoremf1imaen 7904 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 7905 The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.)
(𝐴 ≈ ∅ ↔ 𝐴 = ∅)

Theoremensn1 7906 A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
𝐴 ∈ V       {𝐴} ≈ 1𝑜

Theoremensn1g 7907 A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → {𝐴} ≈ 1𝑜)

Theoremenpr1g 7908 {𝐴, 𝐴} has only one element. (Contributed by FL, 15-Feb-2010.)
(𝐴𝑉 → {𝐴, 𝐴} ≈ 1𝑜)

Theoremen1 7909* A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
(𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})

Theoremen1b 7910 A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
(𝐴 ≈ 1𝑜𝐴 = { 𝐴})

Theoremreuen1 7911* Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝐴 𝜑 ↔ {𝑥𝐴𝜑} ≈ 1𝑜)

Theoremeuen1 7912 Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃!𝑥𝜑 ↔ {𝑥𝜑} ≈ 1𝑜)

Theoremeuen1b 7913* Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥𝐴)

Theoremen1uniel 7914 A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
(𝑆 ≈ 1𝑜 𝑆𝑆)

Theorem2dom 7915* A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.)
(2𝑜𝐴 → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦)

Theoremfundmen 7916 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 7917 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 7918 A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.)
((Rel 𝐴𝐴𝑉) → 𝐴𝐴)

Theoremfndmeng 7919 A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.)
((𝐹 Fn 𝐴𝐴𝐶) → 𝐴𝐹)

Theoremmapsnen 7920 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 7921 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 7922 Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.)
((𝐴𝐶𝐵𝐷) → {𝐴} ≈ {𝐵})

Theoremsnfi 7923 A singleton is finite. (Contributed by NM, 4-Nov-2002.)
{𝐴} ∈ Fin

Theoremfiprc 7924 The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
Fin ∉ V

Theoremunen 7925 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 7926 Any subset of a countable set is countable. (Contributed by Thierry Arnoux, 31-Jan-2017.)
((𝐴𝐵𝐵 ≼ ω) → 𝐴 ≼ ω)

Theoremdifsnen 7927 All decrements of a set are equinumerous. (Contributed by Stefan O'Rear, 19-Feb-2015.)
((𝑋𝑉𝐴𝑋𝐵𝑋) → (𝑋 ∖ {𝐴}) ≈ (𝑋 ∖ {𝐵}))

Theoremdomdifsn 7928 Dominance over a set with one element removed. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
(𝐴𝐵𝐴 ≼ (𝐵 ∖ {𝐶}))

Theoremxpsnen 7929 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 7930 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 7931 One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(𝐴𝑉 → (𝐴 × 1𝑜) ≈ 𝐴)

Theoremendisj 7932* 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) = ∅)

Theoremundom 7933 Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
(((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼ (𝐵𝐷))

Theoremxpcomf1o 7934* 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 7935* Composition with the bijection of xpcomf1o 7934 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 7936 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 7937 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 7938 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 7939 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 7940 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 7941 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 7942 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))

Theoremxpdom3 7943 A set is dominated by its Cartesian product with a nonempty set. Exercise 6 of [Suppes] p. 98. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 × 𝐵))

Theoremxpdom1 7944 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))

Theoremdomunsncan 7945 A singleton cancellation law for dominance. (Contributed by Stefan O'Rear, 19-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.)
𝐴 ∈ V    &   𝐵 ∈ V       ((¬ 𝐴𝑋 ∧ ¬ 𝐵𝑌) → (({𝐴} ∪ 𝑋) ≼ ({𝐵} ∪ 𝑌) ↔ 𝑋𝑌))

Theoremomxpenlem 7946* Lemma for omxpen 7947. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 25-May-2015.)
𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵))

Theoremomxpen 7947 The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ≈ (𝐴 × 𝐵))

Theoremomf1o 7948* Construct an explicit bijection from 𝐴 ·𝑜 𝐵 to 𝐵 ·𝑜 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))    &   𝐺 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥))       ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺𝐹):(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))

Theorempw2f1olem 7949* Lemma for pw2f1o 7950. (Contributed by Mario Carneiro, 6-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑊)    &   (𝜑𝐵𝐶)       (𝜑 → ((𝑆 ∈ 𝒫 𝐴𝐺 = (𝑧𝐴 ↦ if(𝑧𝑆, 𝐶, 𝐵))) ↔ (𝐺 ∈ ({𝐵, 𝐶} ↑𝑚 𝐴) ∧ 𝑆 = (𝐺 “ {𝐶}))))

Theorempw2f1o 7950* The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑊)    &   (𝜑𝐵𝐶)    &   𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝑧𝐴 ↦ if(𝑧𝑥, 𝐶, 𝐵)))       (𝜑𝐹:𝒫 𝐴1-1-onto→({𝐵, 𝐶} ↑𝑚 𝐴))

Theorempw2eng 7951 The power set of a set is equinumerous to set exponentiation with a base of ordinal 2𝑜. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 1-Jul-2015.)
(𝐴𝑉 → 𝒫 𝐴 ≈ (2𝑜𝑚 𝐴))

Theorempw2en 7952 The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. This is Metamath 100 proof #52. (Contributed by NM, 29-Jan-2004.) (Proof shortened by Mario Carneiro, 1-Jul-2015.)
𝐴 ∈ V       𝒫 𝐴 ≈ (2𝑜𝑚 𝐴)

Theoremfopwdom 7953 Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) (Revised by AV, 18-Sep-2021.)
((𝐹𝑉𝐹:𝐴onto𝐵) → 𝒫 𝐵 ≼ 𝒫 𝐴)

Theoremenfixsn 7954* Given two equipollent sets, a bijection can always be chosen which fixes a single point. (Contributed by Stefan O'Rear, 9-Jul-2015.)
((𝐴𝑋𝐵𝑌𝑋𝑌) → ∃𝑓(𝑓:𝑋1-1-onto𝑌 ∧ (𝑓𝐴) = 𝐵))

2.4.24  Schroeder-Bernstein Theorem

Theoremsbthlem1 7955* Lemma for sbth 7965. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}        𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))

Theoremsbthlem2 7956* Lemma for sbth 7965. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)

Theoremsbthlem3 7957* Lemma for sbth 7965. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))

Theoremsbthlem4 7958* Lemma for sbth 7965. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))

Theoremsbthlem5 7959* Lemma for sbth 7965. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)

Theoremsbthlem6 7960* Lemma for sbth 7965. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)

Theoremsbthlem7 7961* Lemma for sbth 7965. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)

Theoremsbthlem8 7962* Lemma for sbth 7965. (Contributed by NM, 27-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)

Theoremsbthlem9 7963* Lemma for sbth 7965. (Contributed by NM, 28-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))       ((𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)

Theoremsbthlem10 7964* Lemma for sbth 7965. (Contributed by NM, 28-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))    &   𝐵 ∈ V       ((𝐴𝐵𝐵𝐴) → 𝐴𝐵)

Theoremsbth 7965 Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set 𝐴 is smaller (has lower cardinality) than 𝐵 and vice-versa, then 𝐴 and 𝐵 are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 7955 through sbthlem10 7964; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 7964. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.)
((𝐴𝐵𝐵𝐴) → 𝐴𝐵)

Theoremsbthb 7966 Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)
((𝐴𝐵𝐵𝐴) ↔ 𝐴𝐵)

Theoremsbthcl 7967 Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
≈ = ( ≼ ∩ ≼ )

Theoremdfsdom2 7968 Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
≺ = ( ≼ ∖ ≼ )

Theorembrsdom2 7969 Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))

Theoremsdomnsym 7970 Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)
(𝐴𝐵 → ¬ 𝐵𝐴)

Theoremdomnsym 7971 Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.)
(𝐴𝐵 → ¬ 𝐵𝐴)

Theorem0domg 7972 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉 → ∅ ≼ 𝐴)

Theoremdom0 7973 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)

Theorem0sdomg 7974 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.)
(𝐴𝑉 → (∅ ≺ 𝐴𝐴 ≠ ∅))

Theorem0dom 7975 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ∅ ≼ 𝐴

Theorem0sdom 7976 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
𝐴 ∈ V       (∅ ≺ 𝐴𝐴 ≠ ∅)

Theoremsdom0 7977 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.)
¬ 𝐴 ≺ ∅

Theoremsdomdomtr 7978 Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremsdomentr 7979 Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremdomsdomtr 7980 Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremensdomtr 7981 Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremsdomirr 7982 Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
¬ 𝐴𝐴

Theoremsdomtr 7983 Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Theoremsdomn2lp 7984 Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.)
¬ (𝐴𝐵𝐵𝐴)

Theoremenen1 7985 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))

Theoremenen2 7986 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))

Theoremdomen1 7987 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))

Theoremdomen2 7988 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))

Theoremsdomen1 7989 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))

Theoremsdomen2 7990 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))

Theoremdomtriord 7991 Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremsdomel 7992 Strict dominance implies ordinal membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))

Theoremsdomdif 7993 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)

Theoremonsdominel 7994 An ordinal with more elements of some type is larger. (Contributed by Stefan O'Rear, 2-Nov-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐴𝐶) ≺ (𝐵𝐶)) → 𝐴𝐵)

Theoremdomunsn 7995 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)

Theoremfodomr 7996* There exists a mapping from a set onto any (nonempty) set that it dominates. (Contributed by NM, 23-Mar-2006.)
((∅ ≺ 𝐵𝐵𝐴) → ∃𝑓 𝑓:𝐴onto𝐵)

Theorempwdom 7997 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵)

Theoremcanth2 7998 Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 6508. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
𝐴 ∈ V       𝐴 ≺ 𝒫 𝐴

Theoremcanth2g 7999 Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.)
(𝐴𝑉𝐴 ≺ 𝒫 𝐴)

Theorem2pwuninel 8000 The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.)
¬ 𝒫 𝒫 𝐴𝐴

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