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Theorem List for Metamath Proof Explorer - 8901-9000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem0sdomgOLD 8901 Obsolete version of 0sdomg 8900 as of 29-Nov-2024. (Contributed by NM, 23-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∅ ≺ 𝐴𝐴 ≠ ∅))
 
Theorem0dom 8902 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ∅ ≼ 𝐴
 
Theorem0sdom 8903 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
𝐴 ∈ V       (∅ ≺ 𝐴𝐴 ≠ ∅)
 
Theoremsdom0 8904 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) Avoid ax-pow 5289, ax-un 7597. (Revised by BTernaryTau, 29-Nov-2024.)
¬ 𝐴 ≺ ∅
 
Theoremsdom0OLD 8905 Obsolete version of sdom0 8904 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝐴 ≺ ∅
 
Theoremsdomdomtr 8906 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 8907 Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomsdomtr 8908 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 8909 Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsdomirr 8910 Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
¬ 𝐴𝐴
 
Theoremsdomtr 8911 Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsdomn2lp 8912 Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.)
¬ (𝐴𝐵𝐵𝐴)
 
Theoremenen1 8913 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremenen2 8914 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremdomen1 8915 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremdomen2 8916 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremsdomen1 8917 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremsdomen2 8918 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremdomtriord 8919 Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremsdomel 8920 For ordinals, strict dominance implies membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
 
Theoremsdomdif 8921 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)
 
Theoremonsdominel 8922 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 8923 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
 
Theoremfodomr 8924* 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 8925 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵)
 
Theoremcanth2 8926 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 7238. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
𝐴 ∈ V       𝐴 ≺ 𝒫 𝐴
 
Theoremcanth2g 8927 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 8928 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 8929 No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
(𝐴𝑉 → 𝒫 𝒫 𝐴𝐴)
 
Theoremdisjen 8930 A stronger form of pwuninel 8100. We can use pwuninel 8100, 2pwuninel 8928 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 8931* Existence version of disjen 8930. (Contributed by Mario Carneiro, 7-Feb-2015.)
((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
 
Theoremdomss2 8932 A corollary of disjenex 8931. 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 8933* A corollary of disjenex 8931. 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 8934* Weakening of domssex2 8933 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))
 
2.4.27  Equinumerosity (cont.)
 
Theoremxpf1o 8935* 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 8936 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 8937 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 8938 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 8939 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 8940* Lemma for xpmapen 8941. (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 8941 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 8942 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 8943 A cardinal power with exponent 2 is equivalent to a Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 17-Jul-2022.)
(𝐴𝑉 → (𝐴m 2o) ≈ (𝐴 × 𝐴))
 
Theoremmapdom2 8944 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 8945 Set exponentiation dominates the base. (Contributed by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 17-Jul-2022.)
((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
 
Theorempwen 8946 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 8947* 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 8948 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 8949 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
Lim 𝐴       (𝐴𝑉𝐴 ≈ suc 𝐴)
 
Theoremlimensuc 8950 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴)
 
Theoreminfensuc 8951 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)
 
2.4.28  Finite sets
 
Theoremdif1enlem 8952 Lemma for rexdif1en 8953 and dif1en 8954. (Contributed by BTernaryTau, 18-Aug-2024.)
((𝐹𝑉𝑀 ∈ ω ∧ 𝐹:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝐹𝑀)}) ≈ 𝑀)
 
Theoremrexdif1en 8953* If a set is equinumerous to a nonzero finite ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
 
Theoremdif1en 8954 If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. For a proof with fewer symbols using ax-pow 5289, see dif1enALT 9059. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5289. (Revised by BTernaryTau, 26-Aug-2024.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
Theoremfindcard 8955* 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 8956* 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.) Avoid ax-pow 5289. (Revised by BTernaryTau, 26-Aug-2024.)
(𝑥 = ∅ → (𝜑𝜓))    &   (𝑥 = 𝑦 → (𝜑𝜒))    &   (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))    &   (𝑥 = 𝐴 → (𝜑𝜏))    &   𝜓    &   (𝑦 ∈ Fin → (𝜒𝜃))       (𝐴 ∈ Fin → 𝜏)
 
Theoremfindcard2s 8957* Variation of findcard2 8956 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 8958* Deduction version of findcard2 8956. (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)
 
Theoremnnfi 8959 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) Avoid ax-pow 5289. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)
 
Theorempssnn 8960* 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.) Avoid ax-pow 5289. (Revised by BTernaryTau, 31-Jul-2024.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥𝐴 𝐵𝑥)
 
Theoremssnnfi 8961 A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.) (Proof shortened by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
 
TheoremssnnfiOLD 8962 Obsolete version of ssnnfi 8961 as of 23-Sep-2024. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
 
Theorem0fin 8963 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
∅ ∈ Fin
 
Theoremunfi 8964 The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.) Avoid ax-pow 5289. (Revised by BTernaryTau, 7-Aug-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
 
Theoremssfi 8965 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. For a shorter proof using ax-pow 5289, see ssfiALT 8966. (Contributed by NM, 24-Jun-1998.) Avoid ax-pow 5289. (Revised by BTernaryTau, 12-Aug-2024.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
 
TheoremssfiALT 8966 Shorter proof of ssfi 8965 using ax-pow 5289. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
 
Theoremimafi 8967 Images of finite sets are finite. For a shorter proof using ax-pow 5289, see imafiALT 9121. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5289. (Revised by BTernaryTau, 7-Sep-2024.)
((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)
 
Theorempwfir 8968 If the power set of a set is finite, then the set itself is finite. (Contributed by BTernaryTau, 7-Sep-2024.)
(𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
 
Theorempwfilem 8969* Lemma for pwfi 8970. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5289. (Revised by BTernaryTau, 7-Sep-2024.)
𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))       (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
 
Theorempwfi 8970 The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5289. (Revised by BTernaryTau, 7-Sep-2024.)
(𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
 
Theoremdiffi 8971 If 𝐴 is finite, (𝐴𝐵) is finite. (Contributed by FL, 3-Aug-2009.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)
 
Theoremcnvfi 8972 If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5289. (Revised by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → 𝐴 ∈ Fin)
 
Theoremfnfi 8973 A version of fnex 7102 for finite sets that does not require Replacement or Power Sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
((𝐹 Fn 𝐴𝐴 ∈ Fin) → 𝐹 ∈ Fin)
 
Theoremf1oenfi 8974 If the domain of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1oeng 8768). (Contributed by BTernaryTau, 8-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1oenfirn 8975 If the range of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. (Contributed by BTernaryTau, 9-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1-onto𝐵) → 𝐴𝐵)
 
Theoremf1domfi 8976 If the codomain of a one-to-one function is finite, then the function's domain is dominated by its codomain. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1domg 8769). (Contributed by BTernaryTau, 25-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐹:𝐴1-1𝐵) → 𝐴𝐵)
 
Theoremf1domfi2 8977 If the domain of a one-to-one function is finite, then the function's domain is dominated by its codomain when the latter is a set. This theorem is proved without using the Axiom of Power Sets (unlike f1dom2g 8766). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵𝑉𝐹:𝐴1-1𝐵) → 𝐴𝐵)
 
Theoremenreffi 8978 Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg 8781). (Contributed by BTernaryTau, 8-Sep-2024.)
(𝐴 ∈ Fin → 𝐴𝐴)
 
Theoremensymfib 8979 Symmetry of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike ensymb 8797). (Contributed by BTernaryTau, 9-Sep-2024.)
(𝐴 ∈ Fin → (𝐴𝐵𝐵𝐴))
 
Theorementrfil 8980 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8801). (Contributed by BTernaryTau, 10-Sep-2024.)
((𝐴 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremenfii 8981 A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5289. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴𝐵) → 𝐴 ∈ Fin)
 
Theoremenfi 8982 Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5289, see enfiALT 8983. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5289. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
 
TheoremenfiALT 8983 Shorter proof of enfi 8982 using ax-pow 5289. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
 
Theoremdomfi 8984 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)
 
Theorementrfi 8985 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8801). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theorementrfir 8986 Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 8801). (Contributed by BTernaryTau, 23-Sep-2024.)
((𝐶 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomtrfil 8987 Transitivity of dominance relation when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domtr 8802). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomtrfi 8988 Transitivity of dominance relation when 𝐵 is finite, proved without using the Axiom of Power Sets (unlike domtr 8802). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomtrfir 8989 Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 8802). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐶 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremf1imaenfi 8990 If a function is one-to-one, then the image of a finite subset of its domain under it is equinumerous to the subset. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1imaeng 8809). (Contributed by BTernaryTau, 29-Sep-2024.)
((𝐹:𝐴1-1𝐵𝐶𝐴𝐶 ∈ Fin) → (𝐹𝐶) ≈ 𝐶)
 
Theoremssdomfi 8991 A finite set dominates its subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8795). (Contributed by BTernaryTau, 12-Nov-2024.)
(𝐵 ∈ Fin → (𝐴𝐵𝐴𝐵))
 
Theoremssdomfi2 8992 A set dominates its finite subsets, proved without using the Axiom of Power Sets (unlike ssdomg 8795). (Contributed by BTernaryTau, 24-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐵𝑉𝐴𝐵) → 𝐴𝐵)
 
Theoremsbthfilem 8993* Lemma for sbthfi 8994. (Contributed by BTernaryTau, 4-Nov-2024.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}    &   𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))    &   𝐵 ∈ V       ((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵)
 
Theoremsbthfi 8994 Schroeder-Bernstein Theorem for finite sets, proved without using the Axiom of Power Sets (unlike sbth 8889). (Contributed by BTernaryTau, 4-Nov-2024.)
((𝐵 ∈ Fin ∧ 𝐴𝐵𝐵𝐴) → 𝐴𝐵)
 
Theoremdomnsymfi 8995 If a set dominates a finite set, it cannot also be strictly dominated by the finite set. This theorem is proved without using the Axiom of Power Sets (unlike domnsym 8895). (Contributed by BTernaryTau, 22-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴𝐵) → ¬ 𝐵𝐴)
 
Theoremsdomdomtrfi 8996 Transitivity of strict dominance and dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike sdomdomtr 8906). (Contributed by BTernaryTau, 25-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomsdomtrfi 8997 Transitivity of dominance and strict dominance when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domsdomtr 8908). (Contributed by BTernaryTau, 25-Nov-2024.)
((𝐴 ∈ Fin ∧ 𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsucdom2 8998 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.) Avoid ax-pow 5289. (Revised by BTernaryTau, 4-Dec-2024.)
(𝐴𝐵 → suc 𝐴𝐵)
 
2.4.29  Pigeonhole Principle
 
Theoremphplem1 8999 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.) Avoid ax-pow 5289. (Revised by BTernaryTau, 23-Sep-2024.)
((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
 
Theoremphplem2 9000 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.) Avoid ax-pow 5289. (Revised by BTernaryTau, 4-Nov-2024.)
𝐴 ∈ V       ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ≈ suc 𝐵𝐴𝐵))
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