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Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempw2f1olem 9101* Lemma for pw2f1o 9102. (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 9102* 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 9103 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 9104 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 9105 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 9106* 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))
 
Theoremsucdom2OLD 9107 Obsolete version of sucdom2 9231 as of 4-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → suc 𝐴𝐵)
 
2.4.30  Schroeder-Bernstein Theorem
 
Theoremsbthlem1 9108* Lemma for sbth 9118. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}        𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
 
Theoremsbthlem2 9109* Lemma for sbth 9118. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
 
Theoremsbthlem3 9110* Lemma for sbth 9118. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
 
Theoremsbthlem4 9111* Lemma for sbth 9118. (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 9112* Lemma for sbth 9118. (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 9113* Lemma for sbth 9118. (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 9114* Lemma for sbth 9118. (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 9115* Lemma for sbth 9118. (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 9116* Lemma for sbth 9118. (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 9117* Lemma for sbth 9118. (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 9118 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. The theorem can also be proved from the axiom of choice and the linear order of the cardinal numbers, but our development does not provide the linear order of cardinal numbers until much later and in ways that depend on Schroeder-Bernstein.

The main proof consists of lemmas sbthlem1 9108 through sbthlem10 9117; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 9117. 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. In the Intuitionistic Logic Explorer (ILE) the Schroeder-Bernstein Theorem has been proven equivalent to the law of the excluded middle (LEM), and in ILE the LEM is not accepted as necessarily true; see https://us.metamath.org/ileuni/exmidsbth.html 9117. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.)

((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
 
Theoremsbthb 9119 Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)
((𝐴𝐵𝐵𝐴) ↔ 𝐴𝐵)
 
Theoremsbthcl 9120 Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
≈ = ( ≼ ∩ ≼ )
 
Theoremdfsdom2 9121 Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
≺ = ( ≼ ∖ ≼ )
 
Theorembrsdom2 9122 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 9123 Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)
(𝐴𝐵 → ¬ 𝐵𝐴)
 
Theoremdomnsym 9124 Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.)
(𝐴𝐵 → ¬ 𝐵𝐴)
 
Theorem0domg 9125 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴𝑉 → ∅ ≼ 𝐴)
 
Theorem0domgOLD 9126 Obsolete version of 0domg 9125 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → ∅ ≼ 𝐴)
 
Theoremdom0 9127 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)
 
Theoremdom0OLD 9128 Obsolete version of dom0 9127 as of 29-Nov-2024. (Contributed by NM, 22-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)
 
Theorem0sdomg 9129 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.)
(𝐴𝑉 → (∅ ≺ 𝐴𝐴 ≠ ∅))
 
Theorem0sdomgOLD 9130 Obsolete version of 0sdomg 9129 as of 29-Nov-2024. (Contributed by NM, 23-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝑉 → (∅ ≺ 𝐴𝐴 ≠ ∅))
 
Theorem0dom 9131 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ∅ ≼ 𝐴
 
Theorem0sdom 9132 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
𝐴 ∈ V       (∅ ≺ 𝐴𝐴 ≠ ∅)
 
Theoremsdom0 9133 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) Avoid ax-pow 5365, ax-un 7741. (Revised by BTernaryTau, 29-Nov-2024.)
¬ 𝐴 ≺ ∅
 
Theoremsdom0OLD 9134 Obsolete version of sdom0 9133 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ 𝐴 ≺ ∅
 
Theoremsdomdomtr 9135 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 9136 Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomsdomtr 9137 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 9138 Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsdomirr 9139 Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
¬ 𝐴𝐴
 
Theoremsdomtr 9140 Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsdomn2lp 9141 Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.)
¬ (𝐴𝐵𝐵𝐴)
 
Theoremenen1 9142 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremenen2 9143 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremdomen1 9144 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremdomen2 9145 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremsdomen1 9146 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremsdomen2 9147 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremdomtriord 9148 Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremsdomel 9149 For ordinals, strict dominance implies membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
 
Theoremsdomdif 9150 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)
 
Theoremonsdominel 9151 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 9152 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
 
Theoremfodomr 9153* 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 9154 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵)
 
Theoremcanth2 9155 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 7372. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
𝐴 ∈ V       𝐴 ≺ 𝒫 𝐴
 
Theoremcanth2g 9156 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 9157 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 9158 No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
(𝐴𝑉 → 𝒫 𝒫 𝐴𝐴)
 
Theoremdisjen 9159 A stronger form of pwuninel 8281. We can use pwuninel 8281, 2pwuninel 9157 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 9160* Existence version of disjen 9159. (Contributed by Mario Carneiro, 7-Feb-2015.)
((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
 
Theoremdomss2 9161 A corollary of disjenex 9160. 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 9162* A corollary of disjenex 9160. 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 9163* Weakening of domssex2 9162 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))
 
2.4.31  Equinumerosity (cont.)
 
Theoremxpf1o 9164* 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 9165 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 9166 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 9167 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 9168 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 9169* Lemma for xpmapen 9170. (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 9170 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 9171 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 9172 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 9173 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 9174 Set exponentiation dominates the base. (Contributed by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 17-Jul-2022.)
((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
 
Theorempwen 9175 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 9176* 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 9177 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 9178 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
Lim 𝐴       (𝐴𝑉𝐴 ≈ suc 𝐴)
 
Theoremlimensuc 9179 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴)
 
Theoreminfensuc 9180 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)
 
2.4.32  Finite sets
 
Theoremdif1enlem 9181 Lemma for rexdif1en 9183 and dif1en 9185. (Contributed by BTernaryTau, 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to avoid ax-un 7741. (Revised by BTernaryTau, 5-Jan-2025.)
(((𝐹𝑉𝐴𝑊𝑀 ∈ On) ∧ 𝐹:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝐹𝑀)}) ≈ 𝑀)
 
Theoremdif1enlemOLD 9182 Obsolete version of dif1enlem 9181 as of 5-Jan-2025. (Contributed by BTernaryTau, 18-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐹𝑉𝑀 ∈ ω ∧ 𝐹:𝐴1-1-onto→suc 𝑀) → (𝐴 ∖ {(𝐹𝑀)}) ≈ 𝑀)
 
Theoremrexdif1en 9183* If a set is equinumerous to a nonzero 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.) Generalize to all ordinals and avoid ax-un 7741. (Revised by BTernaryTau, 5-Jan-2025.)
((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
 
Theoremrexdif1enOLD 9184* Obsolete version of rexdif1en 9183 as of 5-Jan-2025. (Contributed by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
 
Theoremdif1en 9185 If a set 𝐴 is equinumerous to the successor of an ordinal 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.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.)
((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
Theoremdif1ennn 9186 If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. See also dif1ennnALT 9302. (Contributed by BTernaryTau, 6-Jan-2025.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
Theoremdif1enOLD 9187 Obsolete version of dif1en 9185 as of 6-Jan-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 
Theoremfindcard 9188* 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 9189* 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 5365. (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 9190* Variation of findcard2 9189 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 9191* Deduction version of findcard2 9189. (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 9192 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 23-Sep-2024.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)
 
Theorempssnn 9193* 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 5365. (Revised by BTernaryTau, 31-Jul-2024.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ∃𝑥𝐴 𝐵𝑥)
 
Theoremssnnfi 9194 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 9195 Obsolete version of ssnnfi 9194 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 9196 The empty set is finite. (Contributed by FL, 14-Jul-2008.)
∅ ∈ Fin
 
Theoremunfi 9197 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 5365. (Revised by BTernaryTau, 7-Aug-2024.)
((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
 
Theoremssfi 9198 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. For a shorter proof using ax-pow 5365, see ssfiALT 9199. (Contributed by NM, 24-Jun-1998.) Avoid ax-pow 5365. (Revised by BTernaryTau, 12-Aug-2024.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
 
TheoremssfiALT 9199 Shorter proof of ssfi 9198 using ax-pow 5365. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
 
Theoremimafi 9200 Images of finite sets are finite. For a shorter proof using ax-pow 5365, see imafiALT 9371. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5365. (Revised by BTernaryTau, 7-Sep-2024.)
((Fun 𝐹𝑋 ∈ Fin) → (𝐹𝑋) ∈ Fin)
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