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Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsdomen1 8101 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))

Theoremsdomen2 8102 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))

Theoremdomtriord 8103 Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremsdomel 8104 Strict dominance implies ordinal membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))

Theoremsdomdif 8105 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)

Theoremonsdominel 8106 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 8107 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)

Theoremfodomr 8108* 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 8109 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵)

Theoremcanth2 8110 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 6605. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
𝐴 ∈ V       𝐴 ≺ 𝒫 𝐴

Theoremcanth2g 8111 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 8112 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

Theorem2pwne 8113 No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
(𝐴𝑉 → 𝒫 𝒫 𝐴𝐴)

Theoremdisjen 8114 A stronger form of pwuninel 7398. We can use pwuninel 7398, 2pwuninel 8112 to create one or two sets disjoint from a given set 𝐴, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set 𝐵 we can construct a set 𝑥 that is equinumerous to it and disjoint from 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.)
((𝐴𝑉𝐵𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ran 𝐴}) ≈ 𝐵))

Theoremdisjenex 8115* Existence version of disjen 8114. (Contributed by Mario Carneiro, 7-Feb-2015.)
((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))

Theoremdomss2 8116 A corollary of disjenex 8115. If 𝐹 is an injection from 𝐴 to 𝐵 then 𝐺 is a right inverse of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
𝐺 = (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ran 𝐴})))       ((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → (𝐺:𝐵1-1-onto→ran 𝐺𝐴 ⊆ ran 𝐺 ∧ (𝐺𝐹) = ( I ↾ 𝐴)))

Theoremdomssex2 8117* A corollary of disjenex 8115. If 𝐹 is an injection from 𝐴 to 𝐵 then there is a right inverse 𝑔 of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹:𝐴1-1𝐵𝐴𝑉𝐵𝑊) → ∃𝑔(𝑔:𝐵1-1→V ∧ (𝑔𝐹) = ( I ↾ 𝐴)))

Theoremdomssex 8118* Weakening of domssex 8118 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
(𝐴𝐵 → ∃𝑥(𝐴𝑥𝐵𝑥))

2.4.25  Equinumerosity (cont.)

Theoremxpf1o 8119* 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 8120 Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐶𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))

Theoremmapen 8121 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))

Theoremmapdom1 8122 Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
(𝐴𝐵 → (𝐴𝑚 𝐶) ≼ (𝐵𝑚 𝐶))

Theoremmapxpen 8123 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 8124* Lemma for xpmapen 8125. (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 8125 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))

Theoremmapunen 8126 Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
(((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ≈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))

Theoremmap2xp 8127 A cardinal power with exponent 2 is equivalent to a Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.)
(𝐴𝑉 → (𝐴𝑚 2𝑜) ≈ (𝐴 × 𝐴))

Theoremmapdom2 8128 Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴𝐵 ∧ ¬ (𝐴 = ∅ ∧ 𝐶 = ∅)) → (𝐶𝑚 𝐴) ≼ (𝐶𝑚 𝐵))

Theoremmapdom3 8129 Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴𝑚 𝐵))

Theorempwen 8130 If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.)
(𝐴𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵)

Theoremssenen 8131* 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)})

Theoremlimenpsi 8132 A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
Lim 𝐴       (𝐴𝑉𝐴 ≈ (𝐴 ∖ {∅}))

Theoremlimensuci 8133 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
Lim 𝐴       (𝐴𝑉𝐴 ≈ suc 𝐴)

Theoremlimensuc 8134 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴)

Theoreminfensuc 8135 Any infinite ordinal is equinumerous to its successor. Exercise 7 of [TakeutiZaring] p. 88. Proved without the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 13-Jan-2013.)
((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)

2.4.26  Pigeonhole Principle

Theoremphplem1 8136 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 8137 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 8138 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

Theoremphplem4 8139 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))

Theoremnneneq 8140 Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Theoremphp 8141 Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 8136 through phplem4 8139, nneneq 8140, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)

Theoremphp2 8142 Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵𝐴)

Theoremphp3 8143 Corollary of Pigeonhole Principle. If 𝐴 is finite and 𝐵 is a proper subset of 𝐴, the 𝐵 is strictly less numerous than 𝐴. Stronger version of Corollary 6C of [Enderton] p. 135. (Contributed by NM, 22-Aug-2008.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)

Theoremphp4 8144 Corollary of the Pigeonhole Principle php 8141: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ ω → 𝐴 ≺ suc 𝐴)

Theoremphp5 8145 Corollary of the Pigeonhole Principle php 8141: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ ω → ¬ 𝐴 ≈ suc 𝐴)

Theoremsnnen2o 8146 A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.)
¬ {𝐴} ≈ 2𝑜

2.4.27  Finite sets

Theoremonomeneq 8147 An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))

Theoremonfin 8148 An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ On → (𝐴 ∈ Fin ↔ 𝐴 ∈ ω))

Theoremonfin2 8149 A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.)
ω = (On ∩ Fin)

Theoremnnfi 8150 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)

Theoremnndomo 8151 Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))

Theoremnnsdomo 8152 Cardinal ordering agrees with natural number ordering. (Contributed by NM, 17-Jun-1998.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))

Theoremsucdom2 8153 Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
(𝐴𝐵 → suc 𝐴𝐵)

Theoremsucdom 8154 Strict dominance of a set over a natural number is the same as dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.)
(𝐴 ∈ ω → (𝐴𝐵 ↔ suc 𝐴𝐵))

Theorem0sdom1dom 8155 Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004.)
(∅ ≺ 𝐴 ↔ 1𝑜𝐴)

Theorem1sdom2 8156 Ordinal 1 is strictly dominated by ordinal 2. (Contributed by NM, 4-Apr-2007.)
1𝑜 ≺ 2𝑜

Theoremsdom1 8157 A set has less than one member iff it is empty. (Contributed by Stefan O'Rear, 28-Oct-2014.)
(𝐴 ≺ 1𝑜𝐴 = ∅)

Theoremmodom 8158 Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
(∃*𝑥𝜑 ↔ {𝑥𝜑} ≼ 1𝑜)

Theoremmodom2 8159* Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.)
(∃*𝑥 𝑥𝐴𝐴 ≼ 1𝑜)

Theorem1sdom 8160* A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8026.) (Contributed by Mario Carneiro, 12-Jan-2013.)
(𝐴𝑉 → (1𝑜𝐴 ↔ ∃𝑥𝐴𝑦𝐴 ¬ 𝑥 = 𝑦))

Theoremunxpdomlem1 8161* Lemma for unxpdom 8164. (Trivial substitution proof.) (Contributed by Mario Carneiro, 13-Jan-2013.)
𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)    &   𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)       (𝑧 ∈ (𝑎𝑏) → (𝐹𝑧) = if(𝑧𝑎, ⟨𝑧, if(𝑧 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑧 = 𝑡, 𝑛, 𝑚), 𝑧⟩))

Theoremunxpdomlem2 8162* Lemma for unxpdom 8164. (Contributed by Mario Carneiro, 13-Jan-2013.)
𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)    &   𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)    &   (𝜑𝑤 ∈ (𝑎𝑏))    &   (𝜑 → ¬ 𝑚 = 𝑛)    &   (𝜑 → ¬ 𝑠 = 𝑡)       ((𝜑 ∧ (𝑧𝑎 ∧ ¬ 𝑤𝑎)) → ¬ (𝐹𝑧) = (𝐹𝑤))

Theoremunxpdomlem3 8163* Lemma for unxpdom 8164. (Contributed by Mario Carneiro, 13-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2014.)
𝐹 = (𝑥 ∈ (𝑎𝑏) ↦ 𝐺)    &   𝐺 = if(𝑥𝑎, ⟨𝑥, if(𝑥 = 𝑚, 𝑡, 𝑠)⟩, ⟨if(𝑥 = 𝑡, 𝑛, 𝑚), 𝑥⟩)       ((1𝑜𝑎 ∧ 1𝑜𝑏) → (𝑎𝑏) ≼ (𝑎 × 𝑏))

Theoremunxpdom 8164 Cartesian product dominates union for sets with cardinality greater than 1. Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 13-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
((1𝑜𝐴 ∧ 1𝑜𝐵) → (𝐴𝐵) ≼ (𝐴 × 𝐵))

Theoremunxpdom2 8165 Corollary of unxpdom 8164. (Contributed by NM, 16-Sep-2004.)
((1𝑜𝐴𝐵𝐴) → (𝐴𝐵) ≼ (𝐴 × 𝐴))

Theoremsucxpdom 8166 Cartesian product dominates successor for set with cardinality greater than 1. Proposition 10.38 of [TakeutiZaring] p. 93 (but generalized to arbitrary sets, not just ordinals). (Contributed by NM, 3-Sep-2004.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
(1𝑜𝐴 → suc 𝐴 ≼ (𝐴 × 𝐴))

Theorempssinf 8167 A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
((𝐴𝐵𝐴𝐵) → ¬ 𝐵 ∈ Fin)

Theoremfisseneq 8168 A finite set is equal to its subset if they are equinumerous. (Contributed by FL, 11-Aug-2008.)
((𝐵 ∈ Fin ∧ 𝐴𝐵𝐴𝐵) → 𝐴 = 𝐵)

Theoremominf 8169 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136. (Contributed by NM, 2-Jun-1998.)
¬ ω ∈ Fin

Theoremisinf 8170* Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.)
𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))

Theoremfineqvlem 8171 Lemma for fineqv 8172. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
((𝐴𝑉 ∧ ¬ 𝐴 ∈ Fin) → ω ≼ 𝒫 𝒫 𝐴)

Theoremfineqv 8172 If the Axiom of Infinity is denied, then all sets are finite (which implies the Axiom of Choice). (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 3-Jan-2015.)
(¬ ω ∈ V ↔ Fin = V)

Theoremenfi 8173 Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.)
(𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))

Theoremenfii 8174 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.)
((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)

Theorempssnn 8175* A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥𝐴 𝐵𝑥)

Theoremssnnfi 8176 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵 ∈ Fin)

Theoremssfi 8177 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. (Contributed by NM, 24-Jun-1998.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)

Theoremdomfi 8178 A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)

Theoremxpfir 8179 The components of a nonempty finite Cartesian product are finite. (Contributed by Paul Chapman, 11-Apr-2009.) (Proof shortened by Mario Carneiro, 29-Apr-2015.)
(((𝐴 × 𝐵) ∈ Fin ∧ (𝐴 × 𝐵) ≠ ∅) → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))

Theoremssfid 8180 A subset of a finite set is finite, deduction version of ssfi 8177. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)       (𝜑𝐵 ∈ Fin)

Theoreminfi 8181 The intersection of two sets is finite if one of them is. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremrabfi 8182* A restricted class built from a finite set is finite. (Contributed by Thierry Arnoux, 14-Feb-2017.)
(𝐴 ∈ Fin → {𝑥𝐴𝜑} ∈ Fin)

Theoremfinresfin 8183 The restriction of a finite set is finite. (Contributed by Alexander van der Vekens, 3-Jan-2018.)
(𝐸 ∈ Fin → (𝐸𝐵) ∈ Fin)

Theoremf1finf1o 8184 Any injection from one finite set to another of equal size must be a bijection. (Contributed by Jeff Madsen, 5-Jun-2010.) (Revised by Mario Carneiro, 27-Feb-2014.)
((𝐴𝐵𝐵 ∈ Fin) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto𝐵))

Theorem0fin 8185 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
∅ ∈ Fin

Theoremnfielex 8186* If a class is not finite, it contains at least one element. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝐴 ∈ Fin → ∃𝑥 𝑥𝐴)

Theoremen1eqsn 8187 A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.)
((𝐴𝐵𝐵 ≈ 1𝑜) → 𝐵 = {𝐴})

Theoremen1eqsnbi 8188 A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 19270. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.)
(𝐴𝐵 → (𝐵 ≈ 1𝑜𝐵 = {𝐴}))

Theoremdiffi 8189 If 𝐴 is finite, (𝐴𝐵) is finite. (Contributed by FL, 3-Aug-2009.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremdif1en 8190 If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Theoremenp1ilem 8191 Lemma for uses of enp1i 8192. (Contributed by Mario Carneiro, 5-Jan-2016.)
𝑇 = ({𝑥} ∪ 𝑆)       (𝑥𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆𝐴 = 𝑇))

Theoremenp1i 8192* Proof induction for en2i 7990 and related theorems. (Contributed by Mario Carneiro, 5-Jan-2016.)
𝑀 ∈ ω    &   𝑁 = suc 𝑀    &   ((𝐴 ∖ {𝑥}) ≈ 𝑀𝜑)    &   (𝑥𝐴 → (𝜑𝜓))       (𝐴𝑁 → ∃𝑥𝜓)

Theoremen2 8193* A set equinumerous to ordinal 2 is a pair. (Contributed by Mario Carneiro, 5-Jan-2016.)
(𝐴 ≈ 2𝑜 → ∃𝑥𝑦 𝐴 = {𝑥, 𝑦})

Theoremen3 8194* A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
(𝐴 ≈ 3𝑜 → ∃𝑥𝑦𝑧 𝐴 = {𝑥, 𝑦, 𝑧})

Theoremen4 8195* A set equinumerous to ordinal 4 is a quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
(𝐴 ≈ 4𝑜 → ∃𝑥𝑦𝑧𝑤 𝐴 = ({𝑥, 𝑦} ∪ {𝑧, 𝑤}))

Theoremfindcard 8196* Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))    &   (𝑥 = 𝑦 → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ Fin → (∀𝑧𝑦 𝜒𝜃))       (𝐴 ∈ Fin → 𝜏)

Theoremfindcard2 8197* Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ Fin → (𝜒𝜃))       (𝐴 ∈ Fin → 𝜏)

Theoremfindcard2s 8198* Variation of findcard2 8197 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝜒𝜃))       (𝐴 ∈ Fin → 𝜏)

Theoremfindcard2d 8199* Deduction version of findcard2 8197. (Contributed by SO, 16-Jul-2018.)
(𝑥 = ∅ → (𝜓𝜒))    &   (𝑥 = 𝑦 → (𝜓𝜃))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓𝜏))    &   (𝑥 = 𝐴 → (𝜓𝜂))    &   (𝜑𝜒)    &   ((𝜑 ∧ (𝑦𝐴𝑧 ∈ (𝐴𝑦))) → (𝜃𝜏))    &   (𝜑𝐴 ∈ Fin)       (𝜑𝜂)

Theoremfindcard3 8200* Schema for strong induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on any proper subset. The result is then proven to be true for all finite sets. (Contributed by Mario Carneiro, 13-Dec-2013.)
(𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   (𝑦 ∈ Fin → (∀𝑥(𝑥𝑦𝜑) → 𝜒))       (𝐴 ∈ Fin → 𝜏)

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