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Type | Label | Description |
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Statement | ||
Theorem | ac6s4 9901* | 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 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)) | ||
Theorem | ac6s5 9902* | 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 ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∃𝑓∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) | ||
Theorem | ac8 9903* | 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.) |
⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) | ||
Theorem | ac9s 9904* | 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 9309). (Contributed by NM, 29-Sep-2006.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ X𝑥 ∈ 𝐴 𝐵 ≠ ∅) | ||
Theorem | numthcor 9905* | Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) | ||
Theorem | weth 9906* | 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.) |
⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 We 𝐴) | ||
Theorem | zorn2lem1 9907* | Lemma for zorn2 9917. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) & ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} & ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} ⇒ ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝐹‘𝑥) ∈ 𝐷) | ||
Theorem | zorn2lem2 9908* | Lemma for zorn2 9917. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) & ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} & ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} ⇒ ⊢ ((𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅)) → (𝑦 ∈ 𝑥 → (𝐹‘𝑦)𝑅(𝐹‘𝑥))) | ||
Theorem | zorn2lem3 9909* | Lemma for zorn2 9917. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) & ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} & ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} ⇒ ⊢ ((𝑅 Po 𝐴 ∧ (𝑥 ∈ On ∧ (𝑤 We 𝐴 ∧ 𝐷 ≠ ∅))) → (𝑦 ∈ 𝑥 → ¬ (𝐹‘𝑥) = (𝐹‘𝑦))) | ||
Theorem | zorn2lem4 9910* | Lemma for zorn2 9917. (Contributed by NM, 3-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) & ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} & ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} ⇒ ⊢ ((𝑅 Po 𝐴 ∧ 𝑤 We 𝐴) → ∃𝑥 ∈ On 𝐷 = ∅) | ||
Theorem | zorn2lem5 9911* | Lemma for zorn2 9917. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) & ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} & ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} & ⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ⇒ ⊢ (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → (𝐹 “ 𝑥) ⊆ 𝐴) | ||
Theorem | zorn2lem6 9912* | Lemma for zorn2 9917. (Contributed by NM, 4-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) & ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} & ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} & ⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ⇒ ⊢ (𝑅 Po 𝐴 → (((𝑤 We 𝐴 ∧ 𝑥 ∈ On) ∧ ∀𝑦 ∈ 𝑥 𝐻 ≠ ∅) → 𝑅 Or (𝐹 “ 𝑥))) | ||
Theorem | zorn2lem7 9913* | Lemma for zorn2 9917. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐹 = recs((𝑓 ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑤𝑣))) & ⊢ 𝐶 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ ran 𝑓 𝑔𝑅𝑧} & ⊢ 𝐷 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑥)𝑔𝑅𝑧} & ⊢ 𝐻 = {𝑧 ∈ 𝐴 ∣ ∀𝑔 ∈ (𝐹 “ 𝑦)𝑔𝑅𝑧} ⇒ ⊢ ((𝐴 ∈ dom card ∧ 𝑅 Po 𝐴 ∧ ∀𝑠((𝑠 ⊆ 𝐴 ∧ 𝑅 Or 𝑠) → ∃𝑎 ∈ 𝐴 ∀𝑟 ∈ 𝑠 (𝑟𝑅𝑎 ∨ 𝑟 = 𝑎))) → ∃𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 ¬ 𝑎𝑅𝑏) | ||
Theorem | zorn2g 9914* | Zorn's Lemma of [Monk1] p. 117. This version of zorn2 9917 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((𝑤 ⊆ 𝐴 ∧ 𝑅 Or 𝑤) → ∃𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝑤 (𝑧𝑅𝑥 ∨ 𝑧 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) | ||
Theorem | zorng 9915* | 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 9918 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((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | zornn0g 9916* | Variant of Zorn's lemma zorng 9915 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((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | zorn2 9917* | 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 9907 through zorn2lem7 9913; this final piece mainly changes bound variables to eliminate the hypotheses of zorn2lem7 9913. (Contributed by NM, 6-Apr-1997.) (Revised by Mario Carneiro, 9-May-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑤((𝑤 ⊆ 𝐴 ∧ 𝑅 Or 𝑤) → ∃𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝑤 (𝑧𝑅𝑥 ∨ 𝑧 = 𝑥))) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) | ||
Theorem | zorn 9918* | 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 9917 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | zornn0 9919* | Variant of Zorn's lemma zorn 9918 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((𝑧 ⊆ 𝐴 ∧ 𝑧 ≠ ∅ ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | ttukeylem1 9920* | Lemma for ttukey 9929. Expand out the property of being an element of a property of finite character. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) ⇒ ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴)) | ||
Theorem | ttukeylem2 9921* | Lemma for ttukey 9929. A property of finite character is closed under subsets. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) ⇒ ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ⊆ 𝐶)) → 𝐷 ∈ 𝐴) | ||
Theorem | ttukeylem3 9922* | Lemma for ttukey 9929. (Contributed by Mario Carneiro, 11-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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) → (𝐺‘𝐶) = if(𝐶 = ∪ 𝐶, if(𝐶 = ∅, 𝐵, ∪ (𝐺 “ 𝐶)), ((𝐺‘∪ 𝐶) ∪ if(((𝐺‘∪ 𝐶) ∪ {(𝐹‘∪ 𝐶)}) ∈ 𝐴, {(𝐹‘∪ 𝐶)}, ∅)))) | ||
Theorem | ttukeylem4 9923* | Lemma for ttukey 9929. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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) | ||
Theorem | ttukeylem5 9924* | Lemma for ttukey 9929. The 𝐺 function forms a (transfinitely long) chain of inclusions. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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 ∧ 𝐶 ⊆ 𝐷)) → (𝐺‘𝐶) ⊆ (𝐺‘𝐷)) | ||
Theorem | ttukeylem6 9925* | Lemma for ttukey 9929. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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 ∈ suc (card‘(∪ 𝐴 ∖ 𝐵))) → (𝐺‘𝐶) ∈ 𝐴) | ||
Theorem | ttukeylem7 9926* | Lemma for ttukey 9929. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝜑 → 𝐹:(card‘(∪ 𝐴 ∖ 𝐵))–1-1-onto→(∪ 𝐴 ∖ 𝐵)) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ (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 → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) | ||
Theorem | ttukey2g 9927* | The Teichmüller-Tukey Lemma ttukey 9929 with a slightly stronger conclusion: we can set up the maximal element of 𝐴 so that it also contains some given 𝐵 ∈ 𝐴 as a subset. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐵 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) | ||
Theorem | ttukeyg 9928* | The Teichmüller-Tukey Lemma ttukey 9929 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(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | ttukey 9929* | 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 𝑥 ∈ 𝐴 iff every finite subset of 𝑥 is in 𝐴. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | ||
Theorem | axdclem 9930* | Lemma for axdc 9932. (Contributed by Mario Carneiro, 25-Jan-2013.) |
⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) ⇒ ⊢ ((∀𝑦 ∈ 𝒫 dom 𝑥(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) ∧ ran 𝑥 ⊆ dom 𝑥 ∧ ∃𝑧(𝐹‘𝐾)𝑥𝑧) → (𝐾 ∈ ω → (𝐹‘𝐾)𝑥(𝐹‘suc 𝐾))) | ||
Theorem | axdclem2 9931* | Lemma for axdc 9932. 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 (𝐹‘𝑥) (where 𝑥 is the value from the previous iteration). (Contributed by Mario Carneiro, 25-Jan-2013.) |
⊢ 𝐹 = (rec((𝑦 ∈ V ↦ (𝑔‘{𝑧 ∣ 𝑦𝑥𝑧})), 𝑠) ↾ ω) ⇒ ⊢ (∃𝑧 𝑠𝑥𝑧 → (ran 𝑥 ⊆ dom 𝑥 → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))) | ||
Theorem | axdc 9932* | This theorem derives ax-dc 9857 using ax-ac 9870 and ax-inf 9090. 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.) |
⊢ ((∃𝑦∃𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) | ||
Theorem | fodom 9933 | An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 9885. AC is not needed for finite sets - see fodomfi 8786. See also fodomnum 9472. (Contributed by NM, 23-Jul-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴) | ||
Theorem | fodomg 9934 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
⊢ (𝐴 ∈ 𝐶 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | ||
Theorem | dmct 9935 | The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω) | ||
Theorem | rnct 9936 | The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → ran 𝐴 ≼ ω) | ||
Theorem | fodomb 9937* | 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:𝐴–onto→𝐵) ↔ (∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴)) | ||
Theorem | wdomac 9938 | 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 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌) | ||
Theorem | brdom3 9939* | Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)) | ||
Theorem | brdom5 9940* | An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)) | ||
Theorem | brdom4 9941* | An equivalence to a dominance relation. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 𝑥𝑓𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑦𝑓𝑥)) | ||
Theorem | brdom7disj 9942* | 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 & ⊢ (𝐴 ∩ 𝐵) = ∅ ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦 ∈ 𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)) | ||
Theorem | brdom6disj 9943* | An equivalence to a dominance relation for disjoint sets. (Contributed by NM, 5-Apr-2007.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ (𝐴 ∩ 𝐵) = ∅ ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓(∀𝑥 ∈ 𝐵 ∃*𝑦{𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 {𝑦, 𝑥} ∈ 𝑓)) | ||
Theorem | fin71ac 9944 | 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 | ||
Theorem | imadomg 9945 | 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.) |
⊢ (𝐴 ∈ 𝐵 → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴)) | ||
Theorem | fimact 9946 | The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.) |
⊢ ((𝐴 ≼ ω ∧ Fun 𝐹) → (𝐹 “ 𝐴) ≼ ω) | ||
Theorem | fnrndomg 9947 | The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.) |
⊢ (𝐴 ∈ 𝐵 → (𝐹 Fn 𝐴 → ran 𝐹 ≼ 𝐴)) | ||
Theorem | fnct 9948 | If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≼ ω) → 𝐹 ≼ ω) | ||
Theorem | mptct 9949* | A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | ||
Theorem | iunfo 9950* | 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 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⇒ ⊢ (2nd ↾ 𝑇):𝑇–onto→∪ 𝑥 ∈ 𝐴 𝐵 | ||
Theorem | iundom2g 9951* | 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 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) & ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) ⇒ ⊢ (𝜑 → 𝑇 ≼ (𝐴 × 𝐶)) | ||
Theorem | iundomg 9952* | 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 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) & ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 (𝐶 ↑m 𝐵) ∈ AC 𝐴) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≼ 𝐶) & ⊢ (𝜑 → (𝐴 × 𝐶) ∈ AC ∪ 𝑥 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ (𝐴 × 𝐶)) | ||
Theorem | iundom 9953* | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ≼ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ≼ (𝐴 × 𝐵)) | ||
Theorem | unidom 9954* | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ 𝐵) → ∪ 𝐴 ≼ (𝐴 × 𝐵)) | ||
Theorem | uniimadom 9955* | 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 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) | ||
Theorem | uniimadomf 9956* | An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 9955 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) |
⊢ Ⅎ𝑥𝐹 & ⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) | ||
Theorem | cardval 9957* | The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 9409 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 ∣ 𝑥 ≈ 𝐴} | ||
Theorem | cardid 9958 | 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 | ||
Theorem | cardidg 9959 | Any set is equinumerous to its cardinal number. Closed theorem form of cardid 9958. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ∈ 𝐵 → (card‘𝐴) ≈ 𝐴) | ||
Theorem | cardidd 9960 | Any set is equinumerous to its cardinal number. Deduction form of cardid 9958. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → (card‘𝐴) ≈ 𝐴) | ||
Theorem | cardf 9961 | 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 | ||
Theorem | carden 9962 |
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 13698 and the finite-set-only hashen 13697.
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 3770). 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 9313). (Contributed by NM, 22-Oct-2003.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵)) | ||
Theorem | cardeq0 9963 | Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.) |
⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) | ||
Theorem | unsnen 9964 | Equinumerosity of a set with a new element added. (Contributed by NM, 7-Nov-2008.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∪ {𝐵}) ≈ suc (card‘𝐴)) | ||
Theorem | carddom 9965 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵)) | ||
Theorem | cardsdom 9966 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) | ||
Theorem | domtri 9967 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) | ||
Theorem | entric 9968 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵 ∨ 𝐵 ≺ 𝐴)) | ||
Theorem | entri2 9969 | Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≺ 𝐴)) | ||
Theorem | entri3 9970 | 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.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≼ 𝐴)) | ||
Theorem | sdomsdomcard 9971 | A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.) |
⊢ (𝐴 ≺ 𝐵 ↔ 𝐴 ≺ (card‘𝐵)) | ||
Theorem | canth3 9972 | 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.) |
⊢ (𝐴 ∈ 𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴)) | ||
Theorem | infxpidm 9973 | 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 9429. (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) |
⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴) | ||
Theorem | ondomon 9974* | 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 8997. (Contributed by NM, 7-Nov-2003.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) | ||
Theorem | cardmin 9975* | The smallest ordinal that strictly dominates a set is a cardinal. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 20-Sep-2014.) |
⊢ (𝐴 ∈ 𝑉 → (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) | ||
Theorem | ficard 9976 | A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ (card‘𝐴) ∈ ω)) | ||
Theorem | infinf 9977 | Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴)) | ||
Theorem | unirnfdomd 9978 | 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.) |
⊢ (𝜑 → 𝐹:𝑇⟶Fin) & ⊢ (𝜑 → ¬ 𝑇 ∈ Fin) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∪ ran 𝐹 ≼ 𝑇) | ||
Theorem | konigthlem 9979* | Lemma for konigth 9980. (Contributed by Mario Carneiro, 22-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) & ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖) & ⊢ 𝐷 = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))) & ⊢ 𝐸 = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖)) ⇒ ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃) | ||
Theorem | konigth 9980* | Konig's Theorem. If 𝑚(𝑖) ≺ 𝑛(𝑖) for all 𝑖 ∈ 𝐴, then Σ𝑖 ∈ 𝐴𝑚(𝑖) ≺ ∏𝑖 ∈ 𝐴𝑛(𝑖), where the sums and products stand in for disjoint union and infinite cartesian product. The version here is proven with 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 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) & ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖) ⇒ ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃) | ||
Theorem | alephsucpw 9981 | The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10087 or gchaleph2 10083.) (Contributed by NM, 27-Aug-2005.) |
⊢ (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) | ||
Theorem | aleph1 9982 | 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) ≼ (2o ↑m (ℵ‘∅)) | ||
Theorem | alephval2 9983* | 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 ∣ ∀𝑦 ∈ 𝐴 (ℵ‘𝑦) ≺ 𝑥}) | ||
Theorem | dominfac 9984 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 9870. See dominf 9856 for a version proved from ax-cc 9846. (Contributed by NM, 25-Mar-2007.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) | ||
Theorem | iunctb 9985* | The countable union of countable sets is countable (indexed union version of unictb 9986). (Contributed by Mario Carneiro, 18-Jan-2014.) |
⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) | ||
Theorem | unictb 9986* | The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 9985 for indexed union version. (Contributed by NM, 26-Mar-2006.) |
⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ω) → ∪ 𝐴 ≼ ω) | ||
Theorem | infmap 9987* | 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 ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | ||
Theorem | alephadd 9988 | 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) ⊔ (ℵ‘𝐵)) ≈ ((ℵ‘𝐴) ∪ (ℵ‘𝐵)) | ||
Theorem | alephmul 9989 | 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))) | ||
Theorem | alephexp1 9990 | 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) ∧ 𝐴 ⊆ 𝐵) → ((ℵ‘𝐴) ↑m (ℵ‘𝐵)) ≈ (2o ↑m (ℵ‘𝐵))) | ||
Theorem | alephsuc3 9991* | An alternate representation of a successor aleph. Compare alephsuc 9483 and alephsuc2 9495. Equality can be obtained by taking the card of the right-hand side then using alephcard 9485 and carden 9962. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) | ||
Theorem | alephexp2 9992* | An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 9990 (which works if the base is less than or equal to the exponent) and infmap 9987 (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 → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}) | ||
Theorem | alephreg 9993 | 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) | ||
Theorem | pwcfsdom 9994* | A corollary of Konig's Theorem konigth 9980. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.) |
⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦))) ⇒ ⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴))) | ||
Theorem | cfpwsdom 9995 | A corollary of Konig's Theorem konigth 9980. Theorem 11.29 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (2o ≼ 𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵 ↑m (ℵ‘𝐴))))) | ||
Theorem | alephom 9996 | From canth2 8659, 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 9980 (in the form of cfpwsdom 9995), (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‘(2o ↑m ω)) ≠ (ℵ‘ω) | ||
Theorem | smobeth 9997 | 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) | ||
Theorem | nd1 9998 | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧) | ||
Theorem | nd2 9999 | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) | ||
Theorem | nd3 10000 | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) |
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