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Type | Label | Description |
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Statement | ||
Theorem | zorn 10401* | 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 10400 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 10402* | Variant of Zorn's lemma zorn 10401 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 10403* | Lemma for ttukey 10412. 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 10404* | Lemma for ttukey 10412. 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 10405* | Lemma for ttukey 10412. (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 10406* | Lemma for ttukey 10412. (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 10407* | Lemma for ttukey 10412. 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 10408* | Lemma for ttukey 10412. (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 10409* | Lemma for ttukey 10412. (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 10410* | The Teichmüller-Tukey Lemma ttukey 10412 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 10411* | The Teichmüller-Tukey Lemma ttukey 10412 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 10412* | 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 10413* | Lemma for axdc 10415. (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 10414* | Lemma for axdc 10415. 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 10415* | This theorem derives ax-dc 10340 using ax-ac 10353 and ax-inf 9532. 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 | fodomg 10416 | An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the axiom of choice ac7g 10368. The axiom of choice is not needed for finite sets, see fodomfi 9227. See also fodomnum 9951. (Contributed by NM, 23-Jul-2004.) (Proof shortened by BJ, 20-May-2024.) |
⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | ||
Theorem | fodom 10417 | An onto function implies dominance of domain over range. (Contributed by NM, 23-Jul-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴) | ||
Theorem | dmct 10418 | The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → dom 𝐴 ≼ ω) | ||
Theorem | rnct 10419 | The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → ran 𝐴 ≼ ω) | ||
Theorem | fodomb 10420* | 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 10421 | 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 10422* | 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 10423* | 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 10424* | 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 10425* | 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 10426* | 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 10427 | 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 10428 | 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 10429 | 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 10430 | 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 10431 | 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 10432* | A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.) |
⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω) | ||
Theorem | iunfo 10433* | 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 10434* | 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 10435* | 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 10436* | 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 10437* | 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 10438* | 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 10439* | An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. This version of uniimadom 10438 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 10440* | The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 9885 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 10441 | 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 10442 | Any set is equinumerous to its cardinal number. Closed theorem form of cardid 10441. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ∈ 𝐵 → (card‘𝐴) ≈ 𝐴) | ||
Theorem | cardidd 10443 | Any set is equinumerous to its cardinal number. Deduction form of cardid 10441. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → (card‘𝐴) ≈ 𝐴) | ||
Theorem | cardf 10444 | 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 10445 |
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 14202 and the finite-set-only hashen 14201.
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 3736). 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 9789). (Contributed by NM, 22-Oct-2003.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵)) | ||
Theorem | cardeq0 10446 | Only the empty set has cardinality zero. (Contributed by NM, 23-Apr-2004.) |
⊢ (𝐴 ∈ 𝑉 → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) | ||
Theorem | unsnen 10447 | 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 10448 | 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 10449 | 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 10450 | 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 10451 | 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 10452 | Trichotomy of dominance and strict dominance. (Contributed by NM, 4-Jan-2004.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≼ 𝐵 ∨ 𝐵 ≺ 𝐴)) | ||
Theorem | entri3 10453 | 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 10454 | A set strictly dominates iff its cardinal strictly dominates. (Contributed by NM, 30-Oct-2003.) |
⊢ (𝐴 ≺ 𝐵 ↔ 𝐴 ≺ (card‘𝐵)) | ||
Theorem | canth3 10455 | 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 10456 | Every infinite class is equinumerous to its Cartesian square. This theorem, which is equivalent to the axiom of choice over ZF, provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. This is a corollary of infxpen 9908 (used via infxpidm2 9911). (Contributed by NM, 17-Sep-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) |
⊢ (ω ≼ 𝐴 → (𝐴 × 𝐴) ≈ 𝐴) | ||
Theorem | ondomon 10457* | The class of ordinals dominated by a given set is an ordinal. Theorem 56 of [Suppes] p. 227. This theorem can be proved without the axiom of choice, see hartogs 9438. (Contributed by NM, 7-Nov-2003.) (Proof modification is discouraged.) Use hartogs 9438 instead. (New usage is discouraged.) |
⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ On ∣ 𝑥 ≼ 𝐴} ∈ On) | ||
Theorem | cardmin 10458* | 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 10459 | 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 10460 | Equivalence between two infiniteness criteria for sets. (Contributed by David Moews, 1-May-2017.) |
⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ Fin ↔ ω ≼ 𝐴)) | ||
Theorem | unirnfdomd 10461 | 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 10462* | Lemma for konigth 10463. (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 10463* | 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 10464 | The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10570 or gchaleph2 10566.) (Contributed by NM, 27-Aug-2005.) |
⊢ (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) | ||
Theorem | aleph1 10465 | 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 10466* | 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 10467 | A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 10353. See dominf 10339 for a version proved from ax-cc 10329. (Contributed by NM, 25-Mar-2007.) |
⊢ 𝐴 ∈ V ⇒ ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) | ||
Theorem | iunctb 10468* | The countable union of countable sets is countable (indexed union version of unictb 10469). (Contributed by Mario Carneiro, 18-Jan-2014.) |
⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝐵 ≼ ω) → ∪ 𝑥 ∈ 𝐴 𝐵 ≼ ω) | ||
Theorem | unictb 10469* | The countable union of countable sets is countable. Theorem 6Q of [Enderton] p. 159. See iunctb 10468 for indexed union version. (Contributed by NM, 26-Mar-2006.) |
⊢ ((𝐴 ≼ ω ∧ ∀𝑥 ∈ 𝐴 𝑥 ≼ ω) → ∪ 𝐴 ≼ ω) | ||
Theorem | infmap 10470* | 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 10471 | 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 10472 | 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 10473 | 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 10474* | An alternate representation of a successor aleph. Compare alephsuc 9962 and alephsuc2 9974. Equality can be obtained by taking the card of the right-hand side then using alephcard 9964 and carden 10445. (Contributed by NM, 23-Oct-2004.) |
⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) ≈ {𝑥 ∈ On ∣ 𝑥 ≈ (ℵ‘𝐴)}) | ||
Theorem | alephexp2 10475* | An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 10473 (which works if the base is less than or equal to the exponent) and infmap 10470 (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 10476 | 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 10477* | A corollary of Konig's Theorem konigth 10463. 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 10478 | A corollary of Konig's Theorem konigth 10463. 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 10479 | From canth2 9032, 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 10463 (in the form of cfpwsdom 10478), (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 10480 | 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 10481 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑦 ∈ 𝑧) | ||
Theorem | nd2 10482 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 1-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑥 𝑧 ∈ 𝑦) | ||
Theorem | nd3 10483 | A lemma for proving conditionless ZFC axioms. (Contributed by NM, 2-Jan-2002.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑥 ∈ 𝑦) | ||
Theorem | nd4 10484 | A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑦 → ¬ ∀𝑧 𝑦 ∈ 𝑥) | ||
Theorem | axextnd 10485 | A version of the Axiom of Extensionality with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 14-Aug-2003.) (New usage is discouraged.) |
⊢ ∃𝑥((𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧) → 𝑦 = 𝑧) | ||
Theorem | axrepndlem1 10486* | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑧 → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))) | ||
Theorem | axrepndlem2 10487 | Lemma for the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 2-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (New usage is discouraged.) |
⊢ (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ ¬ ∀𝑦 𝑦 = 𝑧) → ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(𝑧 ∈ 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑)))) | ||
Theorem | axrepnd 10488 | A version of the Axiom of Replacement with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ ∃𝑥(∃𝑦∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧(∀𝑦 𝑧 ∈ 𝑥 ↔ ∃𝑥(∀𝑧 𝑥 ∈ 𝑦 ∧ ∀𝑦𝜑))) | ||
Theorem | axunndlem1 10489* | Lemma for the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axunnd 10490 | A version of the Axiom of Union with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) |
⊢ ∃𝑥∀𝑦(∃𝑥(𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axpowndlem1 10491 | Lemma for the Axiom of Power Sets with no distinct variable conditions. (Contributed by NM, 4-Jan-2002.) |
⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | ||
Theorem | axpowndlem2 10492* | Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.) (New usage is discouraged.) |
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥))) | ||
Theorem | axpowndlem3 10493* | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 4-Jan-2002.) (Revised by Mario Carneiro, 10-Dec-2016.) (Proof shortened by Wolf Lammen, 10-Jun-2019.) (New usage is discouraged.) |
⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) | ||
Theorem | axpowndlem4 10494 | Lemma for the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.) |
⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (¬ ∀𝑦 𝑦 = 𝑧 → (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)))) | ||
Theorem | axpownd 10495 | A version of the Axiom of Power Sets with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 4-Jan-2002.) (New usage is discouraged.) |
⊢ (¬ 𝑥 = 𝑦 → ∃𝑥∀𝑦(∀𝑥(∃𝑧 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)) | ||
Theorem | axregndlem1 10496 | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 3-Jan-2002.) (New usage is discouraged.) |
⊢ (∀𝑥 𝑥 = 𝑧 → (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))) | ||
Theorem | axregndlem2 10497* | Lemma for the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Mario Carneiro, 10-Dec-2016.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | axregnd 10498 | A version of the Axiom of Regularity with no distinct variable conditions. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 3-Jan-2002.) (Proof shortened by Wolf Lammen, 18-Aug-2019.) (New usage is discouraged.) |
⊢ (𝑥 ∈ 𝑦 → ∃𝑥(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | axinfndlem1 10499* | Lemma for the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.) |
⊢ (∀𝑥 𝑦 ∈ 𝑧 → ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) | ||
Theorem | axinfnd 10500 | A version of the Axiom of Infinity with no distinct variable conditions. (New usage is discouraged.) (Contributed by NM, 5-Jan-2002.) |
⊢ ∃𝑥(𝑦 ∈ 𝑧 → (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
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