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Theorem List for Metamath Proof Explorer - 9701-9800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuniimadom 9701* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.)
𝐴 ∈ V    &   𝐵 ∈ V       ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))

Theoremuniimadomf 9702* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9701 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.)
𝑥𝐹    &   𝐴 ∈ V    &   𝐵 ∈ V       ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))

3.2.3  Cardinal number theorems using Axiom of Choice

Theoremcardval 9703* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 9150 for a simpler version of its value. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝐴 ∈ V       (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴}

Theoremcardid 9704 Any set is equinumerous to its cardinal number. Proposition 10.5 of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 28-Apr-2015.)
𝐴 ∈ V       (card‘𝐴) ≈ 𝐴

Theoremcardidg 9705 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 9704. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 → (card‘𝐴) ≈ 𝐴)

Theoremcardidd 9706 Any set is equinumerous to its cardinal number. Deduction form of cardid 9704. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (card‘𝐴) ≈ 𝐴)

Theoremcardf 9707 The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
card:V⟶On

Theoremcarden 9708 Two sets are equinumerous iff their cardinal numbers are equal. This important theorem expresses the essential concept behind "cardinality" or "size." This theorem appears as Proposition 10.10 of [TakeutiZaring] p. 85, Theorem 7P of [Enderton] p. 197, and Theorem 9 of [Suppes] p. 242 (among others). The Axiom of Choice is required for its proof. Related theorems are hasheni 13453 and the finite-set-only hashen 13452.

This theorem is also known as Hume's Principle. Gottlob Frege's two-volume Grundgesetze der Arithmetik used his Basic Law V to prove this theorem. Unfortunately Basic Law V caused Frege's system to be inconsistent because it was subject to Russell's paradox (see ru 3651). Later scholars have found that Frege primarily used Basic Law V to Hume's Principle. If Basic Law V is replaced by Hume's Principle in Frege's system, much of Frege's work is restored. Grundgesetze der Arithmetik, once Basic Law V is replaced, proves "Frege's theorem" (the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle). See https://plato.stanford.edu/entries/frege-theorem . We take a different approach, using first-order logic and ZFC, to prove the Peano axioms of arithmetic.

The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 9055). (Contributed by NM, 22-Oct-2003.)

((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))

Theoremcardeq0 9709 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))

Theoremunsnen 9710 Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.)
𝐴 ∈ V    &   𝐵 ∈ V       𝐵𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴))

Theoremcarddom 9711 Two sets have the dominance relationship iff their cardinalities have the subset relationship. Equation i of [Quine] p. 232. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))

Theoremcardsdom 9712 Two sets have the strict dominance relationship iff their cardinalities have the membership relationship. Corollary 19.7(2) of [Eisenberg] p. 310. (Contributed by NM, 22-Oct-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))

Theoremdomtri 9713 Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theorementric 9714 Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of [Suppes] p. 242. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐴𝐵𝐵𝐴))

Theorementri2 9715 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐵𝐴))

Theorementri3 9716 Trichotomy of dominance. This theorem is equivalent to the Axiom of Choice. Part of Proposition 4.42(d) of [Mendelson] p. 275. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐵𝐴))

Theoremsdomsdomcard 9717 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
(𝐴𝐵𝐴 ≺ (card‘𝐵))

Theoremcanth3 9718 Cantor's theorem in terms of cardinals. This theorem tells us that no matter how large a cardinal number is, there is a still larger cardinal number. Theorem 18.12 of [Monk1] p. 133. (Contributed by NM, 5-Nov-2003.)
(𝐴𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))

Theoreminfxpidm 9719 The Cartesian product of an infinite set with itself is idempotent. This theorem (which is an AC equivalent) provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This proof follows as a corollary of infxpen 9170. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
(ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)

Theoremondomon 9720* The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227. This theorem can be proved (with a longer proof) without the Axiom of Choice; see hartogs 8738. (Contributed by NM, 7-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)

Theoremcardmin 9721* The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.)
(𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})

Theoremficard 9722 A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐴𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω))

Theoreminfinf 9723 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))

Theoremunirnfdomd 9724 The union of the range of a function from an infinite set into the class of finite sets is dominated by its domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐹:𝑇⟶Fin)    &   (𝜑 → ¬ 𝑇 ∈ Fin)    &   (𝜑𝑇𝑉)       (𝜑 ran 𝐹𝑇)

Theoremkonigthlem 9725* Lemma for konigth 9726. (Contributed by Mario Carneiro, 22-Feb-2013.)
𝐴 ∈ V    &   𝑆 = 𝑖𝐴 (𝑀𝑖)    &   𝑃 = X𝑖𝐴 (𝑁𝑖)    &   𝐷 = (𝑖𝐴 ↦ (𝑎 ∈ (𝑀𝑖) ↦ ((𝑓𝑎)‘𝑖)))    &   𝐸 = (𝑖𝐴 ↦ (𝑒𝑖))       (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → 𝑆𝑃)

Theoremkonigth 9726* Konig's Theorem. If 𝑚(𝑖) ≺ 𝑛(𝑖) for all 𝑖𝐴, then Σ𝑖𝐴𝑚(𝑖) ≺ ∏𝑖𝐴𝑛(𝑖), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with regular unions rather than disjoint unions for convenience, but the version with disjoint unions is clearly a special case of this version. The Axiom of Choice is needed for this proof, but it contains AC as a simple corollary (letting 𝑚(𝑖) = ∅, this theorem says that an infinite cartesian product of nonempty sets is nonempty), so this is an AC equivalent. Theorem 11.26 of [TakeutiZaring] p. 107. (Contributed by Mario Carneiro, 22-Feb-2013.)
𝐴 ∈ V    &   𝑆 = 𝑖𝐴 (𝑀𝑖)    &   𝑃 = X𝑖𝐴 (𝑁𝑖)       (∀𝑖𝐴 (𝑀𝑖) ≺ (𝑁𝑖) → 𝑆𝑃)

Theoremalephsucpw 9727 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9833 or gchaleph2 9829.) (Contributed by NM, 27-Aug-2005.)
(ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴)

Theoremaleph1 9728 The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.)
(ℵ‘1o) ≼ (2o𝑚 (ℵ‘∅))

Theoremalephval2 9729* An alternate way to express the value of the aleph function for nonzero arguments. Theorem 64 of [Suppes] p. 229. (Contributed by NM, 15-Nov-2003.)
((𝐴 ∈ On ∧ ∅ ∈ 𝐴) → (ℵ‘𝐴) = {𝑥 ∈ On ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})

Theoremdominfac 9730 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 9616. See dominf 9602 for a version proved from ax-cc 9592. (Contributed by NM, 25-Mar-2007.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)

3.2.4  Cardinal number arithmetic using Axiom of Choice

Theoremiunctb 9731* The countable union of countable sets is countable (indexed union version of unictb 9732). (Contributed by Mario Carneiro, 18-Jan-2014.)
((𝐴 ≼ ω ∧ ∀𝑥𝐴 𝐵 ≼ ω) → 𝑥𝐴 𝐵 ≼ ω)

Theoremunictb 9732* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 9731 for indexed union version. (Contributed by NM, 26-Mar-2006.)
((𝐴 ≼ ω ∧ ∀𝑥𝐴 𝑥 ≼ ω) → 𝐴 ≼ ω)

Theoreminfmap 9733* An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. (Contributed by NM, 1-Oct-2004.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
((ω ≼ 𝐴𝐵𝐴) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})

Theoremalephadd 9734 The sum of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
((ℵ‘𝐴) +𝑐 (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵))

Theoremalephmul 9735 The product of two alephs is their maximum. Equation 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) × (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)))

Theoremalephexp1 9736 An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
(((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐵)) ≈ (2o𝑚 (ℵ‘𝐵)))

Theoremalephsuc3 9737* An alternate representation of a successor aleph. Compare alephsuc 9224 and alephsuc2 9236. Equality can be obtained by taking the card of the right-hand side then using alephcard 9226 and carden 9708. (Contributed by NM, 23-Oct-2004.)
(𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})

Theoremalephexp2 9738* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9736 (which works if the base is less than or equal to the exponent) and infmap 9733 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.)
(𝐴 ∈ On → (2o𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})

3.2.5  Cofinality using the Axiom of Choice

Theoremalephreg 9739 A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)
(cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)

Theorempwcfsdom 9740* A corollary of Konig's Theorem konigth 9726. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓𝑦)))       (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))

Theoremcfpwsdom 9741 A corollary of Konig's Theorem konigth 9726. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)
𝐵 ∈ V       (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵𝑚 (ℵ‘𝐴)))))

Theoremalephom 9742 From canth2 8401, we know that (ℵ‘0) < (2↑ω), but we cannot prove that (2↑ω) = (ℵ‘1) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement (ℵ‘𝐴) < (2↑ω) is consistent for any ordinal 𝐴). However, we can prove that (2↑ω) is not equal to (ℵ‘ω), nor (ℵ‘(ℵ‘ω)), on cofinality grounds, because by Konig's Theorem konigth 9726 (in the form of cfpwsdom 9741), (2↑ω) has uncountable cofinality, which eliminates limit alephs like (ℵ‘ω). (The first limit aleph that is not eliminated is (ℵ‘(ℵ‘1)), which has cofinality (ℵ‘1).) (Contributed by Mario Carneiro, 21-Mar-2013.)
(card‘(2o𝑚 ω)) ≠ (ℵ‘ω)

Theoremsmobeth 9743 The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as (card‘(𝑅1‘(ω +o 𝐴))), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.)
Smo (card ∘ 𝑅1)

3.3  ZFC Axioms with no distinct variable requirements

Theoremnd1 9744 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)

Theoremnd2 9745 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)

Theoremnd3 9746 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)

Theoremnd4 9747 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦𝑥)

Theoremaxextnd 9748 A version of the Axiom of Extensionality with no distinct variable conditions. (Contributed by NM, 14-Aug-2003.)
𝑥((𝑥𝑦𝑥𝑧) → 𝑦 = 𝑧)

Theoremaxrepndlem1 9749* Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
(¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))

Theoremaxrepndlem2 9750 Lemma for the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
(((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(𝑧𝑥 ↔ ∃𝑥(𝑥𝑦 ∧ ∀𝑦𝜑))))

Theoremaxrepnd 9751 A version of the Axiom of Replacement with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
𝑥(∃𝑦𝑧(𝜑𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧𝑥 ↔ ∃𝑥(∀𝑧 𝑥𝑦 ∧ ∀𝑦𝜑)))

Theoremaxunndlem1 9752* Lemma for the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Theoremaxunnd 9753 A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Theoremaxpowndlem1 9754 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))

Theoremaxpowndlem2 9755* Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.)
(¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))

Theoremaxpowndlem3 9756* Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.)
𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))

Theoremaxpowndlem4 9757 Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
(¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))

Theoremaxpownd 9758 A version of the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.)
𝑥 = 𝑦 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))

Theoremaxregndlem1 9759 Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑥 𝑥 = 𝑧 → (𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦))))

Theoremaxregndlem2 9760* Lemma for the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Theoremaxregnd 9761 A version of the Axiom of Regularity with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
(𝑥𝑦 → ∃𝑥(𝑥𝑦 ∧ ∀𝑧(𝑧𝑥 → ¬ 𝑧𝑦)))

Theoremaxinfndlem1 9762* Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
(∀𝑥 𝑦𝑧 → ∃𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))

Theoremaxinfnd 9763 A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.)
𝑥(𝑦𝑧 → (𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥))))

Theoremaxacndlem1 9764 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Theoremaxacndlem2 9765 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Theoremaxacndlem3 9766 Lemma for the Axiom of Choice with no distinct variable conditions. (Contributed by NM, 3-Jan-2002.)
(∀𝑦 𝑦 = 𝑧 → ∃𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤)))

Theoremaxacndlem4 9767* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremaxacndlem5 9768* Lemma for the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremaxacnd 9769 A version of the Axiom of Choice with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.)
𝑥𝑦𝑧(∀𝑥(𝑦𝑧𝑧𝑤) → ∃𝑤𝑦(∃𝑤((𝑦𝑧𝑧𝑤) ∧ (𝑦𝑤𝑤𝑥)) ↔ 𝑦 = 𝑤))

Theoremzfcndext 9770* Axiom of Extensionality ax-ext 2754, reproved from conditionless ZFC version and predicate calculus. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
(∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦)

Theoremzfcndrep 9771* Axiom of Replacement ax-rep 5006, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
(∀𝑤𝑦𝑧(∀𝑦𝜑𝑧 = 𝑦) → ∃𝑦𝑧(𝑧𝑦 ↔ ∃𝑤(𝑤𝑥 ∧ ∀𝑦𝜑)))

Theoremzfcndun 9772* Axiom of Union ax-un 7226, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
𝑦𝑧(∃𝑤(𝑧𝑤𝑤𝑥) → 𝑧𝑦)

Theoremzfcndpow 9773* Axiom of Power Sets ax-pow 5077, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness" dtru 5082. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremzfcndreg 9774* Axiom of Regularity ax-reg 8786, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
(∃𝑦 𝑦𝑥 → ∃𝑦(𝑦𝑥 ∧ ∀𝑧(𝑧𝑦 → ¬ 𝑧𝑥)))

Theoremzfcndinf 9775* Axiom of Infinity ax-inf 8832, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing theorem el 5081 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)
𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))

Theoremzfcndac 9776* Axiom of Choice ax-ac 9616, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣))

3.4  The Generalized Continuum Hypothesis

3.4.1  Sets satisfying the Generalized Continuum Hypothesis

Syntaxcgch 9777 Extend class notation to include the collection of sets that satisfy the GCH.
class GCH

Definitiondf-gch 9778* Define the collection of "GCH-sets", or sets for which the generalized continuum hypothesis holds. In this language the generalized continuum hypothesis can be expressed as GCH = V. A set 𝑥 satisfies the generalized continuum hypothesis if it is finite or there is no set 𝑦 strictly between 𝑥 and its powerset in cardinality. The continuum hypothesis is equivalent to ω ∈ GCH. (Contributed by Mario Carneiro, 15-May-2015.)
GCH = (Fin ∪ {𝑥 ∣ ∀𝑦 ¬ (𝑥𝑦𝑦 ≺ 𝒫 𝑥)})

Theoremelgch 9779* Elementhood in the collection of GCH-sets. (Contributed by Mario Carneiro, 15-May-2015.)
(𝐴𝑉 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))

Theoremfingch 9780 A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
Fin ⊆ GCH

Theoremgchi 9781 The only GCH-sets which have other sets between it and its power set are finite sets. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 ∈ GCH ∧ 𝐴𝐵𝐵 ≺ 𝒫 𝐴) → 𝐴 ∈ Fin)

Theoremgchen1 9782 If 𝐴𝐵 < 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐴 = 𝐵 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≺ 𝒫 𝐴)) → 𝐴𝐵)

Theoremgchen2 9783 If 𝐴 < 𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → 𝐵 ≈ 𝒫 𝐴)

Theoremgchor 9784 If 𝐴𝐵 ≤ 𝒫 𝐴, and 𝐴 is an infinite GCH-set, then either 𝐴 = 𝐵 or 𝐵 = 𝒫 𝐴 in cardinality. (Contributed by Mario Carneiro, 15-May-2015.)
(((𝐴 ∈ GCH ∧ ¬ 𝐴 ∈ Fin) ∧ (𝐴𝐵𝐵 ≼ 𝒫 𝐴)) → (𝐴𝐵𝐵 ≈ 𝒫 𝐴))

Theoremengch 9785 The property of being a GCH-set is a cardinal invariant. (Contributed by Mario Carneiro, 15-May-2015.)
(𝐴𝐵 → (𝐴 ∈ GCH ↔ 𝐵 ∈ GCH))

Theoremgchdomtri 9786 Under certain conditions, a GCH-set can demonstrate trichotomy of dominance. Lemma for gchac 9838. (Contributed by Mario Carneiro, 15-May-2015.)
((𝐴 ∈ GCH ∧ (𝐴 +𝑐 𝐴) ≈ 𝐴𝐵 ≼ 𝒫 𝐴) → (𝐴𝐵𝐵𝐴))

Theoremfpwwe2cbv 9787* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 3-Jun-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}       𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎𝐴𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧𝑎 [(𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}

Theoremfpwwe2lem1 9788* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}       𝑊 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))

Theoremfpwwe2lem2 9789* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)       (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))

Theoremfpwwe2lem3 9790* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 19-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑋𝑊𝑅)       ((𝜑𝐵𝑋) → ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵)

Theoremfpwwe2lem5 9791* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)       ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)

Theoremfpwwe2lem6 9792* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀𝐵) = (𝑁𝐵))       ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑋𝐶𝑌 ∧ (𝑀𝐶) = (𝑁𝐶)))

Theoremfpwwe2lem7 9793* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑𝐵 ∈ dom 𝑀)    &   (𝜑𝐵 ∈ dom 𝑁)    &   (𝜑 → (𝑀𝐵) = (𝑁𝐵))       ((𝜑𝐶𝑅(𝑀𝐵)) → (𝐶𝑆(𝑁𝐵) ∧ (𝐷𝑅(𝑀𝐵) → (𝐶𝑅𝐷𝐶𝑆𝐷))))

Theoremfpwwe2lem8 9794* Lemma for fpwwe2 9800. Show by induction that the two isometries 𝑀 and 𝑁 agree on their common domain. (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)       (𝜑𝑀 = (𝑁 ↾ dom 𝑀))

Theoremfpwwe2lem9 9795* Lemma for fpwwe2 9800. Given two well-orders 𝑋, 𝑅 and 𝑌, 𝑆 of parts of 𝐴, one is an initial segment of the other. (The 𝑂𝑃 hypothesis is in order to break the symmetry of 𝑋 and 𝑌.) (Contributed by Mario Carneiro, 15-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)    &   𝑀 = OrdIso(𝑅, 𝑋)    &   𝑁 = OrdIso(𝑆, 𝑌)    &   (𝜑 → dom 𝑀 ⊆ dom 𝑁)       (𝜑 → (𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))))

Theoremfpwwe2lem10 9796* Lemma for fpwwe2 9800. Given two well-orders 𝑋, 𝑅 and 𝑌, 𝑆 of parts of 𝐴, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   (𝜑𝑋𝑊𝑅)    &   (𝜑𝑌𝑊𝑆)       (𝜑 → ((𝑋𝑌𝑅 = (𝑆 ∩ (𝑌 × 𝑋))) ∨ (𝑌𝑋𝑆 = (𝑅 ∩ (𝑋 × 𝑌)))))

Theoremfpwwe2lem11 9797* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 15-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))

Theoremfpwwe2lem12 9798* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 18-May-2015.) (Proof shortened by Peter Mazsa, 23-Sep-2022.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑𝑋 ∈ dom 𝑊)

Theoremfpwwe2lem13 9799* Lemma for fpwwe2 9800. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → (𝑋𝐹(𝑊𝑋)) ∈ 𝑋)

Theoremfpwwe2 9800* Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset 𝑋, (𝑊𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 9186. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.)
𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}    &   (𝜑𝐴 ∈ V)    &   ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)    &   𝑋 = dom 𝑊       (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋𝑅 = (𝑊𝑋))))

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