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Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxpdom2 8601 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 8602 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 8603 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 8604 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 8605 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 8606 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 8607* Lemma for omxpen 8608. (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 8608 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 8609* 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 8610* Lemma for pw2f1o 8611. (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 8611* 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 8612 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 8613 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 8614 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 8615* 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))
 
2.4.25  Schroeder-Bernstein Theorem
 
Theoremsbthlem1 8616* Lemma for sbth 8626. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}        𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
 
Theoremsbthlem2 8617* Lemma for sbth 8626. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐷)
 
Theoremsbthlem3 8618* Lemma for sbth 8626. (Contributed by NM, 22-Mar-1998.)
𝐴 ∈ V    &   𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}       (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
 
Theoremsbthlem4 8619* Lemma for sbth 8626. (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 8620* Lemma for sbth 8626. (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 8621* Lemma for sbth 8626. (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 8622* Lemma for sbth 8626. (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 8623* Lemma for sbth 8626. (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 8624* Lemma for sbth 8626. (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 8625* Lemma for sbth 8626. (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 8626 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 8616 through sbthlem10 8625; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 8625. 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 8625. This is Metamath 100 proof #25. (Contributed by NM, 8-Jun-1998.)

((𝐴𝐵𝐵𝐴) → 𝐴𝐵)
 
Theoremsbthb 8627 Schroeder-Bernstein Theorem and its converse. (Contributed by NM, 8-Jun-1998.)
((𝐴𝐵𝐵𝐴) ↔ 𝐴𝐵)
 
Theoremsbthcl 8628 Schroeder-Bernstein Theorem in class form. (Contributed by NM, 28-Mar-1998.)
≈ = ( ≼ ∩ ≼ )
 
Theoremdfsdom2 8629 Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97. (Contributed by NM, 31-Mar-1998.)
≺ = ( ≼ ∖ ≼ )
 
Theorembrsdom2 8630 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 8631 Strict dominance is asymmetric. Theorem 21(ii) of [Suppes] p. 97. (Contributed by NM, 8-Jun-1998.)
(𝐴𝐵 → ¬ 𝐵𝐴)
 
Theoremdomnsym 8632 Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.)
(𝐴𝐵 → ¬ 𝐵𝐴)
 
Theorem0domg 8633 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
(𝐴𝑉 → ∅ ≼ 𝐴)
 
Theoremdom0 8634 A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
(𝐴 ≼ ∅ ↔ 𝐴 = ∅)
 
Theorem0sdomg 8635 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.)
(𝐴𝑉 → (∅ ≺ 𝐴𝐴 ≠ ∅))
 
Theorem0dom 8636 Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐴 ∈ V       ∅ ≼ 𝐴
 
Theorem0sdom 8637 A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
𝐴 ∈ V       (∅ ≺ 𝐴𝐴 ≠ ∅)
 
Theoremsdom0 8638 The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.)
¬ 𝐴 ≺ ∅
 
Theoremsdomdomtr 8639 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 8640 Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremdomsdomtr 8641 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 8642 Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsdomirr 8643 Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
¬ 𝐴𝐴
 
Theoremsdomtr 8644 Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
Theoremsdomn2lp 8645 Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.)
¬ (𝐴𝐵𝐵𝐴)
 
Theoremenen1 8646 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremenen2 8647 Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremdomen1 8648 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremdomen2 8649 Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremsdomen1 8650 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐴𝐶𝐵𝐶))
 
Theoremsdomen2 8651 Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
(𝐴𝐵 → (𝐶𝐴𝐶𝐵))
 
Theoremdomtriord 8652 Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theoremsdomel 8653 Strict dominance implies ordinal membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵𝐴𝐵))
 
Theoremsdomdif 8654 The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
(𝐴𝐵 → (𝐵𝐴) ≠ ∅)
 
Theoremonsdominel 8655 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 8656 Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
(𝐴𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
 
Theoremfodomr 8657* 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 8658 Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
(𝐴𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵)
 
Theoremcanth2 8659 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 7100. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
𝐴 ∈ V       𝐴 ≺ 𝒫 𝐴
 
Theoremcanth2g 8660 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 8661 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 8662 No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
(𝐴𝑉 → 𝒫 𝒫 𝐴𝐴)
 
Theoremdisjen 8663 A stronger form of pwuninel 7932. We can use pwuninel 7932, 2pwuninel 8661 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 8664* Existence version of disjen 8663. (Contributed by Mario Carneiro, 7-Feb-2015.)
((𝐴𝑉𝐵𝑊) → ∃𝑥((𝐴𝑥) = ∅ ∧ 𝑥𝐵))
 
Theoremdomss2 8665 A corollary of disjenex 8664. 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 8666* A corollary of disjenex 8664. 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 8667* Weakening of domssex 8667 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))
 
2.4.26  Equinumerosity (cont.)
 
Theoremxpf1o 8668* 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 8669 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 8670 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 8671 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 8672 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 8673* Lemma for xpmapen 8674. (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 8674 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 8675 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 8676 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 8677 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 8678 Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 17-Jul-2022.)
((𝐴𝑉𝐵𝑊𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
 
Theorempwen 8679 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 8680* 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 8681 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 8682 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
Lim 𝐴       (𝐴𝑉𝐴 ≈ suc 𝐴)
 
Theoremlimensuc 8683 A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
((𝐴𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴)
 
Theoreminfensuc 8684 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)
 
2.4.27  Pigeonhole Principle
 
Theoremphplem1 8685 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 8686 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 8687 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 8688 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 8689 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 8690 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 8685 through phplem4 8688, nneneq 8689, and this final piece of the proof. (Contributed by NM, 29-May-1998.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → ¬ 𝐴𝐵)
 
Theoremphp2 8691 Corollary of Pigeonhole Principle. (Contributed by NM, 31-May-1998.)
((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐵𝐴)
 
Theoremphp3 8692 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 8693 Corollary of the Pigeonhole Principle php 8690: a natural number is strictly dominated by its successor. (Contributed by NM, 26-Jul-2004.)
(𝐴 ∈ ω → 𝐴 ≺ suc 𝐴)
 
Theoremphp5 8694 Corollary of the Pigeonhole Principle php 8690: 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)
 
Theoremphpeqd 8695 Corollary of the Pigeonhole Principle using equality. Strengthening of php 8690 expressed without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.)
(𝜑𝐴 ∈ Fin)    &   (𝜑𝐵𝐴)    &   (𝜑𝐴𝐵)       (𝜑𝐴 = 𝐵)
 
Theoremsnnen2o 8696 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
 
2.4.28  Finite sets
 
Theoremonomeneq 8697 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 8698 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 8699 A set is a natural number iff it is a finite ordinal. (Contributed by Mario Carneiro, 22-Jan-2013.)
ω = (On ∩ Fin)
 
Theoremnnfi 8700 Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.)
(𝐴 ∈ ω → 𝐴 ∈ Fin)
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