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Theorem List for Metamath Proof Explorer - 9901-10000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremac6s4 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       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
 
Theoremac6s5 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       (∀𝑥𝐴 𝐵 ≠ ∅ → ∃𝑓𝑥𝐴 (𝑓𝑥) ∈ 𝐵)
 
Theoremac8 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.)
((∀𝑧𝑥 𝑧 ≠ ∅ ∧ ∀𝑧𝑥𝑤𝑥 (𝑧𝑤 → (𝑧𝑤) = ∅)) → ∃𝑦𝑧𝑥 ∃!𝑣 𝑣 ∈ (𝑧𝑦))
 
Theoremac9s 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       (∀𝑥𝐴 𝐵 ≠ ∅ ↔ X𝑥𝐴 𝐵 ≠ ∅)
 
3.2.2  AC equivalents: well-ordering, Zorn's lemma
 
Theoremnumthcor 9905* Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003.)
(𝐴𝑉 → ∃𝑥 ∈ On 𝐴𝑥)
 
Theoremweth 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.)
(𝐴𝑉 → ∃𝑥 𝑥 We 𝐴)
 
Theoremzorn2lem1 9907* Lemma for zorn2 9917. (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 9908* Lemma for zorn2 9917. (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 9909* Lemma for zorn2 9917. (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 9910* Lemma for zorn2 9917. (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 9911* Lemma for zorn2 9917. (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 9912* Lemma for zorn2 9917. (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 9913* Lemma for zorn2 9917. (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 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((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
 
Theoremzorng 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((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremzornn0g 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((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremzorn2 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((𝑤𝐴𝑅 Or 𝑤) → ∃𝑥𝐴𝑧𝑤 (𝑧𝑅𝑥𝑧 = 𝑥))) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑅𝑦)
 
Theoremzorn 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((𝑧𝐴 ∧ [] Or 𝑧) → 𝑧𝐴) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremzornn0 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((𝑧𝐴𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremttukeylem1 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.)
(𝜑𝐹:(card‘( 𝐴𝐵))–1-1-onto→( 𝐴𝐵))    &   (𝜑𝐵𝐴)    &   (𝜑 → ∀𝑥(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴))       (𝜑 → (𝐶𝐴 ↔ (𝒫 𝐶 ∩ Fin) ⊆ 𝐴))
 
Theoremttukeylem2 9921* Lemma for ttukey 9929. 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 9922* Lemma for ttukey 9929. (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 9923* Lemma for ttukey 9929. (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 9924* Lemma for ttukey 9929. 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 9925* Lemma for ttukey 9929. (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 9926* Lemma for ttukey 9929. (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 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 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 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(𝑥𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥𝐴𝑦𝐴 ¬ 𝑥𝑦)
 
Theoremttukey 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 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 9930* Lemma for axdc 9932. (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 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 (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 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.)
((∃𝑦𝑧 𝑦𝑥𝑧 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
 
Theoremfodom 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:𝐴onto𝐵𝐵𝐴)
 
Theoremfodomg 9934 An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.)
(𝐴𝐶 → (𝐹:𝐴onto𝐵𝐵𝐴))
 
Theoremdmct 9935 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → dom 𝐴 ≼ ω)
 
Theoremrnct 9936 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → ran 𝐴 ≼ ω)
 
Theoremfodomb 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:𝐴onto𝐵) ↔ (∅ ≺ 𝐵𝐵𝐴))
 
Theoremwdomac 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* 𝑌𝑋𝑌)
 
Theorembrdom3 9939* Equivalence to a dominance relation. (Contributed by NM, 27-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
 
Theorembrdom5 9940* An equivalence to a dominance relation. (Contributed by NM, 29-Mar-2007.)
𝐵 ∈ V       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦 𝑥𝑓𝑦 ∧ ∀𝑥𝐴𝑦𝐵 𝑦𝑓𝑥))
 
Theorembrdom4 9941* 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 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    &   (𝐴𝐵) = ∅       (𝐴𝐵 ↔ ∃𝑓(∀𝑥𝐵 ∃*𝑦𝐴 {𝑥, 𝑦} ∈ 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 {𝑦, 𝑥} ∈ 𝑓))
 
Theorembrdom6disj 9943* 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 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
 
Theoremimadomg 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.)
(𝐴𝐵 → (Fun 𝐹 → (𝐹𝐴) ≼ 𝐴))
 
Theoremfimact 9946 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 9947 The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004.)
(𝐴𝐵 → (𝐹 Fn 𝐴 → ran 𝐹𝐴))
 
Theoremfnct 9948 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 9949* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)
 
Theoremiunfo 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 = 𝑥𝐴 ({𝑥} × 𝐵)       (2nd𝑇):𝑇onto 𝑥𝐴 𝐵
 
Theoremiundom2g 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 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶m 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)       (𝜑𝑇 ≼ (𝐴 × 𝐶))
 
Theoremiundomg 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 = 𝑥𝐴 ({𝑥} × 𝐵)    &   (𝜑 𝑥𝐴 (𝐶m 𝐵) ∈ AC 𝐴)    &   (𝜑 → ∀𝑥𝐴 𝐵𝐶)    &   (𝜑 → (𝐴 × 𝐶) ∈ AC 𝑥𝐴 𝐵)       (𝜑 𝑥𝐴 𝐵 ≼ (𝐴 × 𝐶))
 
Theoremiundom 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.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐶𝐵) → 𝑥𝐴 𝐶 ≼ (𝐴 × 𝐵))
 
Theoremunidom 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.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝑥𝐵) → 𝐴 ≼ (𝐴 × 𝐵))
 
Theoremuniimadom 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 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
 
Theoremuniimadomf 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.)
𝑥𝐹    &   𝐴 ∈ V    &   𝐵 ∈ V       ((Fun 𝐹 ∧ ∀𝑥𝐴 (𝐹𝑥) ≼ 𝐵) → (𝐹𝐴) ≼ (𝐴 × 𝐵))
 
3.2.3  Cardinal number theorems using Axiom of Choice
 
Theoremcardval 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 ∣ 𝑥𝐴}
 
Theoremcardid 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
 
Theoremcardidg 9959 Any set is equinumerous to its cardinal number. Closed theorem form of cardid 9958. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 → (card‘𝐴) ≈ 𝐴)
 
Theoremcardidd 9960 Any set is equinumerous to its cardinal number. Deduction form of cardid 9958. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴𝐵)       (𝜑 → (card‘𝐴) ≈ 𝐴)
 
Theoremcardf 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
 
Theoremcarden 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.)

((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
 
Theoremcardeq0 9963 Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.)
(𝐴𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅))
 
Theoremunsnen 9964 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 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.)
((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
 
Theoremcardsdom 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.)
((𝐴𝑉𝐵𝑊) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))
 
Theoremdomtri 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.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 
Theorementric 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.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐴𝐵𝐵𝐴))
 
Theorementri2 9969 Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐵𝐴))
 
Theorementri3 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.)
((𝐴𝑉𝐵𝑊) → (𝐴𝐵𝐵𝐴))
 
Theoremsdomsdomcard 9971 A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.)
(𝐴𝐵𝐴 ≺ (card‘𝐵))
 
Theoremcanth3 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.)
(𝐴𝑉 → (card‘𝐴) ∈ (card‘𝒫 𝐴))
 
Theoreminfxpidm 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)
 
Theoremondomon 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.)
(𝐴𝑉 → {𝑥 ∈ On ∣ 𝑥𝐴} ∈ On)
 
Theoremcardmin 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.)
(𝐴𝑉 → (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})
 
Theoremficard 9976 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 9977 Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.)
(𝐴𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴))
 
Theoremunirnfdomd 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.)
(𝜑𝐹:𝑇⟶Fin)    &   (𝜑 → ¬ 𝑇 ∈ Fin)    &   (𝜑𝑇𝑉)       (𝜑 ran 𝐹𝑇)
 
Theoremkonigthlem 9979* Lemma for konigth 9980. (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 9980* 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 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 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)
 
Theoremaleph1 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) ≼ (2om (ℵ‘∅))
 
Theoremalephval2 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 ∣ ∀𝑦𝐴 (ℵ‘𝑦) ≺ 𝑥})
 
Theoremdominfac 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)
 
3.2.4  Cardinal number arithmetic using Axiom of Choice
 
Theoremiunctb 9985* The countable union of countable sets is countable (indexed union version of unictb 9986). (Contributed by Mario Carneiro, 18-Jan-2014.)
((𝐴 ≼ ω ∧ ∀𝑥𝐴 𝐵 ≼ ω) → 𝑥𝐴 𝐵 ≼ ω)
 
Theoremunictb 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 ∧ ∀𝑥𝐴 𝑥 ≼ ω) → 𝐴 ≼ ω)
 
Theoreminfmap 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 ≼ 𝐴𝐵𝐴) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
 
Theoremalephadd 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))
 
Theoremalephmul 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)))
 
Theoremalephexp1 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) ∧ 𝐴𝐵) → ((ℵ‘𝐴) ↑m (ℵ‘𝐵)) ≈ (2om (ℵ‘𝐵)))
 
Theoremalephsuc3 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)})
 
Theoremalephexp2 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 → (2om (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
 
3.2.5  Cofinality using the Axiom of Choice
 
Theoremalephreg 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)
 
Theorempwcfsdom 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‘(𝑓𝑦)))       (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑m (cf‘(ℵ‘𝐴)))
 
Theoremcfpwsdom 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       (2o𝐵 → (ℵ‘𝐴) ≺ (cf‘(card‘(𝐵m (ℵ‘𝐴)))))
 
Theoremalephom 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‘(2om ω)) ≠ (ℵ‘ω)
 
Theoremsmobeth 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)
 
3.3  ZFC Axioms with no distinct variable requirements
 
Theoremnd1 9998 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦𝑧)
 
Theoremnd2 9999 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 1-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧𝑦)
 
Theoremnd3 10000 A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.)
(∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥𝑦)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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