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Theorem List for Metamath Proof Explorer - 9301-9400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremac6c5 9301* Equivalent of Axiom of Choice. 𝐵 is a collection 𝐵(𝑥) of nonempty sets. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)

Theoremac9 9302* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. (Contributed by Mario Carneiro, 22-Mar-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)

Theoremac6s 9303* Equivalent of Axiom of Choice. Using the Boundedness Axiom bnd2 8753, we derive this strong version of ac6 9299 that doesn't require 𝐵 to be a set. (Contributed by NM, 4-Feb-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6n 9304* Equivalent of Axiom of Choice. Contrapositive of ac6s 9303. (Contributed by NM, 10-Jun-2007.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑓(𝑓:𝐴𝐵 → ∃𝑥𝐴 𝜓) → ∃𝑥𝐴𝑦𝐵 𝜑)

Theoremac6s2 9305* Generalization of the Axiom of Choice to classes. Slightly strengthened version of ac6s3 9306. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 𝜓))

Theoremac6s3 9306* Generalization of the Axiom of Choice to classes. Theorem 10.46 of [TakeutiZaring] p. 97. (Contributed by NM, 3-Nov-2004.)
𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝜑 → ∃𝑓𝑥𝐴 𝜓)

Theoremac6sg 9307* ac6s 9303 with sethood as antecedent. (Contributed by FL, 3-Aug-2009.)
(𝑦 = (𝑓𝑥) → (𝜑𝜓))       (𝐴𝑉 → (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓)))

Theoremac6sf 9308* Version of ac6 9299 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.)
𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremac6s4 9309* Generalization of the Axiom of Choice to proper classes. 𝐵 is a collection 𝐵(𝑥) of nonempty, possible proper classes. (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))

Theoremac6s5 9310* Generalization of the Axiom of Choice to proper classes. 𝐵 is a collection 𝐵(𝑥) of nonempty, possible proper classes. Remark after Theorem 10.46 of [TakeutiZaring] p. 98. (Contributed by NM, 27-Mar-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)

Theoremac8 9311* An Axiom of Choice equivalent. Given a family 𝑥 of mutually disjoint nonempty sets, there exists a set 𝑦 containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))

Theoremac9s 9312* An Axiom of Choice equivalent: the infinite Cartesian product of nonempty classes is nonempty. Axiom of Choice (second form) of [Enderton] p. 55 and its converse. This is a stronger version of the axiom in Enderton, with no existence requirement for the family of classes 𝐵(𝑥) (achieved via the Collection Principle cp 8751). (Contributed by NM, 29-Sep-2006.)
𝐴 ∈ V       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)

3.2.2  AC equivalents: well-ordering, Zorn's lemma

Theoremnumthcor 9313* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)

Theoremweth 9314* Well-ordering theorem: any set 𝐴 can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242. First proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario Carneiro, 5-Jan-2013.)
(𝐴𝑉 → ∃𝑥 𝑥 We 𝐴)

Theoremzorn2lem1 9315* Lemma for zorn2 9325. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝐹𝑥) ∈ 𝐷)

Theoremzorn2lem2 9316* Lemma for zorn2 9325. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅)) → (𝑦𝑥 → (𝐹𝑦)𝑅(𝐹𝑥)))

Theoremzorn2lem3 9317* Lemma for zorn2 9325. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴𝐷 ≠ ∅))) → (𝑦𝑥 → ¬ (𝐹𝑥) = (𝐹𝑦)))

Theoremzorn2lem4 9318* Lemma for zorn2 9325. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}       ((𝑅 Po 𝐴𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅)

Theoremzorn2lem5 9319* Lemma for zorn2 9325. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → (𝐹𝑥) ⊆ 𝐴)

Theoremzorn2lem6 9320* Lemma for zorn2 9325. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       (𝑅 Po 𝐴 → (((𝑤 We 𝐴𝑥 ∈ On) ∧ ∀𝑦𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹𝑥)))

Theoremzorn2lem7 9321* Lemma for zorn2 9325. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐹 = recs((𝑓 ∈ V ↦ (𝑣𝐶𝑢𝐶 ¬ 𝑢𝑤𝑣)))    &   𝐶 = {𝑧𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧}    &   𝐷 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑥)𝑔𝑅𝑧}    &   𝐻 = {𝑧𝐴 ∣ ∀𝑔 ∈ (𝐹𝑦)𝑔𝑅𝑧}       ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑠((𝑠𝐴𝑅 Or 𝑠) → ∃𝑎𝐴𝑟𝑠 (𝑟𝑅𝑎𝑟 = 𝑎))) → ∃𝑎𝐴𝑏𝐴 ¬ 𝑎𝑅𝑏)

Theoremzorn2g 9322* Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9325 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)

Theoremzorng 9323* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. Theorem 6M of [Enderton] p. 151. This version of zorn 9326 avoids the Axiom of Choice by assuming that 𝐴 is well-orderable. (Contributed by NM, 12-Aug-2004.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ ∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzornn0g 9324* Variant of Zorn's lemma zorng 9323 in which , the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Mario Carneiro, 9-May-2015.)
((𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzorn2 9325* Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set 𝐴 (with an ordering relation 𝑅) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zorn2lem1 9315 through zorn2lem7 9321; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 9321. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.)
𝐴 ∈ V       ((𝑅 Po 𝐴 ∧ ∀𝑤((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)

Theoremzorn 9326* Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 9325 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.)
𝐴 ∈ V       (∀𝑧((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremzornn0 9327* Variant of Zorn's lemma zorn 9326 in which , the union of the empty chain, is not required to be an element of 𝐴. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ ∀𝑧((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremttukeylem1 9328* Lemma for ttukey 9337. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))

Theoremttukeylem2 9329* Lemma for ttukey 9337. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       ((𝜑 ∧ (𝐶𝐴𝐷𝐶)) → 𝐷𝐴)

Theoremttukeylem3 9330* Lemma for ttukey 9337. (Contributed by Mario Carneiro, 11-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑𝐶 ∈ On) → (𝐺𝐶) = if(𝐶 = 𝐶, if(𝐶 = ∅, 𝐵, (𝐺𝐶)), ((𝐺 𝐶) ∪ if(((𝐺 𝐶) ∪ {(𝐹 𝐶)}) ∈ 𝐴, {(𝐹 𝐶)}, ∅))))

Theoremttukeylem4 9331* Lemma for ttukey 9337. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       (𝜑 → (𝐺‘∅) = 𝐵)

Theoremttukeylem5 9332* Lemma for ttukey 9337. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑 ∧ (𝐶 ∈ On ∧ 𝐷 ∈ On ∧ 𝐶𝐷)) → (𝐺𝐶) ⊆ (𝐺𝐷))

Theoremttukeylem6 9333* Lemma for ttukey 9337. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       ((𝜑𝐶 ∈ suc (card‘( 𝐴𝐵))) → (𝐺𝐶) ∈ 𝐴)

Theoremttukeylem7 9334* Lemma for ttukey 9337. (Contributed by Mario Carneiro, 15-May-2015.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))    &   𝐺 = recs((𝑧 ∈ V ↦ if(dom 𝑧 = dom 𝑧, if(dom 𝑧 = ∅, 𝐵, ran 𝑧), ((𝑧 dom 𝑧) ∪ if(((𝑧 dom 𝑧) ∪ {(𝐹 dom 𝑧)}) ∈ 𝐴, {(𝐹 dom 𝑧)}, ∅)))))       (𝜑 → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))

Theoremttukey2g 9335* The Teichmüller-Tukey Lemma ttukey 9337 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.)
(( 𝐴 ∈ dom card ∧ 𝐵𝐴 ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴 (𝐵𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥𝑦))

Theoremttukeyg 9336* The Teichmüller-Tukey Lemma ttukey 9337 stated with the "choice" as an antecedent (the hypothesis 𝐴 ∈ dom card says that 𝐴 is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.)
(( 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremttukey 9337* The Teichmüller-Tukey Lemma, an Axiom of Choice equivalent. If 𝐴 is a nonempty collection of finite character, then 𝐴 has a maximal element with respect to inclusion. Here "finite character" means that 𝑥𝐴 iff every finite subset of 𝑥 is in 𝐴. (Contributed by Mario Carneiro, 15-May-2015.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)

Theoremaxdclem 9338* Lemma for axdc 9340. (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)       ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹𝐾)𝑥𝑧) → (𝐾 ∈ ω → (𝐹𝐾)𝑥(𝐹‘suc 𝐾)))

Theoremaxdclem2 9339* Lemma for axdc 9340. Using the full Axiom of Choice, we can construct a choice function 𝑔 on 𝒫 dom 𝑥. From this, we can build a sequence 𝐹 starting at any value 𝑠 ∈ dom 𝑥 by repeatedly applying 𝑔 to the set (𝐹𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.)
𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧𝑦𝑥𝑧})), 𝑠) ↾ ω)       (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))

Theoremaxdc 9340* This theorem derives ax-dc 9265 using ax-ac 9278 and ax-inf 8532. Thus, AC implies DC, but not vice-versa (so that ZFC is strictly stronger than ZF+DC). (New usage is discouraged.) (Contributed by Mario Carneiro, 25-Jan-2013.)
((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))

Theoremfodom 9341 An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 9293. AC is not needed for finite sets - see fodomfi 8236. See also fodomnum 8877. (Contributed by NM, 23-Jul-2004.)
𝐴 ∈ V       (𝐹:𝐴onto𝐵𝐵𝐴)

Theoremfodomg 9342 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵𝐴))

Theoremdmct 9343 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → dom 𝐴 ≼ ω)

Theoremrnct 9344 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → ran 𝐴 ≼ ω)

Theoremfodomb 9345* Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)
((𝐴 ≠ ∅ ∧ ∃𝑓 𝑓:𝐴onto𝐵) ↔ (∅ ≺ 𝐵𝐵𝐴))

Theoremwdomac 9346 When assuming AC, weak and usual dominance coincide. It is not known if this is an AC equivalent. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
(𝑋* 𝑌𝑋𝑌)

Theorembrdom3 9347* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))

Theorembrdom5 9348* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))

Theorembrdom4 9349* An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))

Theorembrdom7disj 9350* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 29-Mar-2007.) (Revised by NM, 16-Jun-2017.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝐴𝐵) = ∅       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))

Theorembrdom6disj 9351* An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝐴𝐵) = ∅       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))

Theoremfin71ac 9352 Once we allow AC, the "strongest" definition of finite set becomes equivalent to the "weakest" and the entire hierarchy collapses. (Contributed by Stefan O'Rear, 29-Oct-2014.)
FinVII = Fin

Theoremimadomg 9353 An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐵 → (Fun 𝐹 → (𝐹𝐴) ≼ 𝐴))

Theoremfimact 9354 The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.)
((𝐴 ≼ ω ∧ Fun 𝐹) → (𝐹𝐴) ≼ ω)

Theoremfnrndomg 9355 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
(𝐴𝐵 → (𝐹 Fn 𝐴 → ran 𝐹𝐴))

Theoremfnct 9356 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)

Theoremmptct 9357* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)

Theoremiunfo 9358* Existence of an onto function from a disjoint union to a union. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 18-Jan-2014.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)       (2nd𝑇):𝑇onto 𝑥𝐴 𝐵

Theoremiundom2g 9359* An upper bound for the cardinality of a disjoint indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶𝑚 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)       (𝜑𝑇 ≼ (𝐴 × 𝐶))

Theoremiundomg 9360* An upper bound for the cardinality of an indexed union, with explicit choice principles. 𝐵 depends on 𝑥 and should be thought of as 𝐵(𝑥). (Contributed by Mario Carneiro, 1-Sep-2015.)
𝑇 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶𝑚 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)    &   (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)       (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))

Theoremiundom 9361* An upper bound for the cardinality of an indexed union. 𝐶 depends on 𝑥 and should be thought of as 𝐶(𝑥). (Contributed by NM, 26-Mar-2006.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ≼ (𝐴 × 𝐵))

Theoremunidom 9362* An upper bound for the cardinality of a union. Theorem 10.47 of [TakeutiZaring] p. 98. (Contributed by NM, 25-Mar-2006.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))

Theoremuniimadom 9363* 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 9364* An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9363 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 9365* The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 8814 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 9366 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 9367 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 9366. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 → (card‘𝐴) ≈ 𝐴)

Theoremcardidd 9368 Any set is equinumerous to its cardinal number. Deduction form of cardid 9366. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (card‘𝐴) ≈ 𝐴)

Theoremcardf 9369 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 9370 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 13131 and the finite-set-only hashen 13130.

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 3432). 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 8755). (Contributed by NM, 22-Oct-2003.)

((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))

Theoremcardeq0 9371 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))

Theoremunsnen 9372 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 9373 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 9374 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 9375 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 9376 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 9377 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐵𝐴))

Theorementri3 9378 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 9379 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
(𝐴𝐵𝐴 ≺ (card‘𝐵))

Theoremcanth3 9380 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 9381 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 8834. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
(ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴)

Theoremondomon 9382* 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 8446. (Contributed by NM, 7-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)

Theoremcardmin 9383* 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 9384 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 9385 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))

Theoremunirnfdomd 9386 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 9387* Lemma for konigth 9388. (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 9388* 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 9389 The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9495 or gchaleph2 9491.) (Contributed by NM, 27-Aug-2005.)
(ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴)

Theoremaleph1 9390 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 9391* 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 9392 A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 9278. See dominf 9264 for a version proved from ax-cc 9254. (Contributed by NM, 25-Mar-2007.)
𝐴 ∈ V       ((𝐴 ≠ ∅ ∧ 𝐴 𝐴) → ω ≼ 𝐴)

3.2.4  Cardinal number arithmetic using Axiom of Choice

Theoremiunctb 9393* The countable union of countable sets is countable (indexed union version of unictb 9394). (Contributed by Mario Carneiro, 18-Jan-2014.)
((𝐴 ≼ ω ∧ ∀𝑥𝐴 𝐵 ≼ ω) → 𝑥𝐴 𝐵 ≼ ω)

Theoremunictb 9394* The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 9393 for indexed union version. (Contributed by NM, 26-Mar-2006.)
((𝐴 ≼ ω ∧ ∀𝑥𝐴 𝑥 ≼ ω) → 𝐴 ≼ ω)

Theoreminfmap 9395* 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 9396 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 9397 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 9398 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 9399* An alternate representation of a successor aleph. Compare alephsuc 8888 and alephsuc2 8900. Equality can be obtained by taking the card of the right-hand side then using alephcard 8890 and carden 9370. (Contributed by NM, 23-Oct-2004.)
(𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)})

Theoremalephexp2 9400* An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9398 (which works if the base is less than or equal to the exponent) and infmap 9395 (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))})

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