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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | kmlem11 10201* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 26-Mar-2004.) |
| ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ ∪ 𝐴) = (𝑧 ∖ ∪ (𝑥 ∖ {𝑧}))) | ||
| Theorem | kmlem12 10202* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 27-Mar-2004.) |
| ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ (∀𝑧 ∈ 𝑥 (𝑧 ∖ ∪ (𝑥 ∖ {𝑧})) ≠ ∅ → (∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) → ∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑦 ∩ ∪ 𝐴))))) | ||
| Theorem | kmlem13 10203* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 5-Apr-2004.) |
| ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))} ⇒ ⊢ (∀𝑥((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥(¬ ∃𝑧 ∈ 𝑥 ∀𝑣 ∈ 𝑧 ∃𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) → ∃𝑦∀𝑧 ∈ 𝑥 (𝑧 ≠ ∅ → ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))) | ||
| Theorem | kmlem14 10204* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
| ⊢ (𝜑 ↔ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣) ∧ 𝑧 ∈ 𝑣))) & ⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))) & ⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ⇒ ⊢ (∃𝑧 ∈ 𝑥 ∀𝑣 ∈ 𝑧 ∃𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) ↔ ∃𝑦∀𝑧∃𝑣∀𝑢(𝑦 ∈ 𝑥 ∧ 𝜑)) | ||
| Theorem | kmlem15 10205* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
| ⊢ (𝜑 ↔ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣) ∧ 𝑧 ∈ 𝑣))) & ⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))) & ⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ⇒ ⊢ ((¬ 𝑦 ∈ 𝑥 ∧ 𝜒) ↔ ∀𝑧∃𝑣∀𝑢(¬ 𝑦 ∈ 𝑥 ∧ 𝜓)) | ||
| Theorem | kmlem16 10206* | Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.) |
| ⊢ (𝜑 ↔ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ 𝑦 ≠ 𝑣) ∧ 𝑧 ∈ 𝑣))) & ⊢ (𝜓 ↔ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))) & ⊢ (𝜒 ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ⇒ ⊢ ((∃𝑧 ∈ 𝑥 ∀𝑣 ∈ 𝑧 ∃𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 ∧ 𝑣 ∈ (𝑧 ∩ 𝑤)) ∨ ∃𝑦(¬ 𝑦 ∈ 𝑥 ∧ 𝜒)) ↔ ∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ 𝜑) ∨ (¬ 𝑦 ∈ 𝑥 ∧ 𝜓))) | ||
| Theorem | dfackm 10207* | Equivalence of the Axiom of Choice and Maes' AC ackm 10505. The proof consists of lemmas kmlem1 10191 through kmlem16 10206 and this final theorem. AC is not used for the proof. Note: bypassing the first step (i.e., replacing dfac5 10169 with biid 261) establishes the AC equivalence shown by Maes' writeup. The left-hand-side AC shown here was chosen because it is shorter to display. (Contributed by NM, 13-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) |
| ⊢ (CHOICE ↔ ∀𝑥∃𝑦∀𝑧∃𝑣∀𝑢((𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑦 → ((𝑣 ∈ 𝑥 ∧ ¬ 𝑦 = 𝑣) ∧ 𝑧 ∈ 𝑣))) ∨ (¬ 𝑦 ∈ 𝑥 ∧ (𝑧 ∈ 𝑥 → ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ∧ ((𝑢 ∈ 𝑧 ∧ 𝑢 ∈ 𝑦) → 𝑢 = 𝑣)))))) | ||
For cardinal arithmetic, we follow [Mendelson] p. 258. Rather than defining operations restricted to cardinal numbers, we use disjoint union df-dju 9941 (⊔) for cardinal addition, Cartesian product df-xp 5691 (×) for cardinal multiplication, and set exponentiation df-map 8868 (↑m) for cardinal exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 10591, carddom 10594, and cardsdom 10595. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available. | ||
| Theorem | undjudom 10208 | Cardinal addition dominates union. (Contributed by NM, 28-Sep-2004.) (Revised by Jim Kingdon, 15-Aug-2023.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ≼ (𝐴 ⊔ 𝐵)) | ||
| Theorem | endjudisj 10209 | Equinumerosity of a disjoint union and a union of two disjoint sets. (Contributed by NM, 5-Apr-2007.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵)) | ||
| Theorem | djuen 10210 | Disjoint unions of equinumerous sets are equinumerous. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷)) | ||
| Theorem | djuenun 10211 | Disjoint union is equinumerous to union for disjoint sets. (Contributed by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 19-Aug-2023.) |
| ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷 ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ∪ 𝐷)) | ||
| Theorem | dju1en 10212 | Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ 𝐴) → (𝐴 ⊔ 1o) ≈ suc 𝐴) | ||
| Theorem | dju1dif 10213 | Adding and subtracting one gives back the original cardinality. Similar to pncan 11514 for cardinalities. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by Jim Kingdon, 20-Aug-2023.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ (𝐴 ⊔ 1o)) → ((𝐴 ⊔ 1o) ∖ {𝐵}) ≈ 𝐴) | ||
| Theorem | dju1p1e2 10214 | 1+1=2 for cardinal number addition, derived from pm54.43 10041 as promised. Theorem *110.643 of Principia Mathematica, vol. II, p. 86, which adds the remark, "The above proposition is occasionally useful." Whitehead and Russell define cardinal addition on collections of all sets equinumerous to 1 and 2 (which for us are proper classes unless we restrict them as in karden 9935), but after applying definitions, our theorem is equivalent. Because we use a disjoint union for cardinal addition (as explained in the comment at the top of this section), we use ≈ instead of =. See dju1p1e2ALT 10215 for a shorter proof that doesn't use pm54.43 10041. (Contributed by NM, 5-Apr-2007.) (Proof modification is discouraged.) |
| ⊢ (1o ⊔ 1o) ≈ 2o | ||
| Theorem | dju1p1e2ALT 10215 | Alternate proof of dju1p1e2 10214. (Contributed by Mario Carneiro, 29-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (1o ⊔ 1o) ≈ 2o | ||
| Theorem | dju0en 10216 | Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ⊔ ∅) ≈ 𝐴) | ||
| Theorem | xp2dju 10217 | Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) | ||
| Theorem | djucomen 10218 | Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ⊔ 𝐵) ≈ (𝐵 ⊔ 𝐴)) | ||
| Theorem | djuassen 10219 | Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ⊔ 𝐵) ⊔ 𝐶) ≈ (𝐴 ⊔ (𝐵 ⊔ 𝐶))) | ||
| Theorem | xpdjuen 10220 | Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 × (𝐵 ⊔ 𝐶)) ≈ ((𝐴 × 𝐵) ⊔ (𝐴 × 𝐶))) | ||
| Theorem | mapdjuen 10221 | Sum of exponents law for cardinal arithmetic. Theorem 6I(4) of [Enderton] p. 142. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ↑m (𝐵 ⊔ 𝐶)) ≈ ((𝐴 ↑m 𝐵) × (𝐴 ↑m 𝐶))) | ||
| Theorem | pwdjuen 10222 | Sum of exponents law for cardinal arithmetic. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 ⊔ 𝐵) ≈ (𝒫 𝐴 × 𝒫 𝐵)) | ||
| Theorem | djudom1 10223 | Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) (Revised by Jim Kingdon, 1-Sep-2023.) |
| ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝐴 ⊔ 𝐶) ≼ (𝐵 ⊔ 𝐶)) | ||
| Theorem | djudom2 10224 | Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝐶 ⊔ 𝐴) ≼ (𝐶 ⊔ 𝐵)) | ||
| Theorem | djudoml 10225 | A set is dominated by its disjoint union with another. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ≼ (𝐴 ⊔ 𝐵)) | ||
| Theorem | djuxpdom 10226 | Cartesian product dominates disjoint union for sets with cardinality greater than 1. Similar to Proposition 10.36 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 18-May-2015.) |
| ⊢ ((1o ≺ 𝐴 ∧ 1o ≺ 𝐵) → (𝐴 ⊔ 𝐵) ≼ (𝐴 × 𝐵)) | ||
| Theorem | djufi 10227 | The disjoint union of two finite sets is finite. (Contributed by NM, 22-Oct-2004.) |
| ⊢ ((𝐴 ≺ ω ∧ 𝐵 ≺ ω) → (𝐴 ⊔ 𝐵) ≺ ω) | ||
| Theorem | cdainflem 10228 | Any partition of omega into two pieces (which may be disjoint) contains an infinite subset. (Contributed by Mario Carneiro, 11-Feb-2013.) |
| ⊢ ((𝐴 ∪ 𝐵) ≈ ω → (𝐴 ≈ ω ∨ 𝐵 ≈ ω)) | ||
| Theorem | djuinf 10229 | A set is infinite iff the cardinal sum with itself is infinite. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (ω ≼ 𝐴 ↔ ω ≼ (𝐴 ⊔ 𝐴)) | ||
| Theorem | infdju1 10230 | An infinite set is equinumerous to itself added with one. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ (ω ≼ 𝐴 → (𝐴 ⊔ 1o) ≈ 𝐴) | ||
| Theorem | pwdju1 10231 | The sum of a powerset with itself is equipotent to the successor powerset. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 (𝐴 ⊔ 1o)) | ||
| Theorem | pwdjuidm 10232 | If the natural numbers inject into 𝐴, then 𝒫 𝐴 is idempotent under cardinal sum. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ (ω ≼ 𝐴 → (𝒫 𝐴 ⊔ 𝒫 𝐴) ≈ 𝒫 𝐴) | ||
| Theorem | djulepw 10233 | If 𝐴 is idempotent under cardinal sum and 𝐵 is dominated by the power set of 𝐴, then so is the cardinal sum of 𝐴 and 𝐵. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ (((𝐴 ⊔ 𝐴) ≈ 𝐴 ∧ 𝐵 ≼ 𝒫 𝐴) → (𝐴 ⊔ 𝐵) ≼ 𝒫 𝐴) | ||
| Theorem | onadju 10234 | The cardinal and ordinal sums are always equinumerous. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Jim Kingdon, 7-Sep-2023.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ≈ (𝐴 ⊔ 𝐵)) | ||
| Theorem | cardadju 10235 | The cardinal sum is equinumerous to an ordinal sum of the cardinals. (Contributed by Mario Carneiro, 6-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) | ||
| Theorem | djunum 10236 | The disjoint union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ⊔ 𝐵) ∈ dom card) | ||
| Theorem | unnum 10237 | The union of two numerable sets is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ∪ 𝐵) ∈ dom card) | ||
| Theorem | nnadju 10238 | The cardinal and ordinal sums of finite ordinals are equal. For a shorter proof using ax-rep 5279, see nnadjuALT 10239. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.) Avoid ax-rep 5279. (Revised by BTernaryTau, 2-Jul-2024.) |
| ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴 ⊔ 𝐵)) = (𝐴 +o 𝐵)) | ||
| Theorem | nnadjuALT 10239 | Shorter proof of nnadju 10238 using ax-rep 5279. (Contributed by Paul Chapman, 11-Apr-2009.) (Revised by Mario Carneiro, 6-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (card‘(𝐴 ⊔ 𝐵)) = (𝐴 +o 𝐵)) | ||
| Theorem | ficardadju 10240 | The disjoint union of finite sets is equinumerous to the ordinal sum of the cardinalities of those sets. (Contributed by BTernaryTau, 3-Jul-2024.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ⊔ 𝐵) ≈ ((card‘𝐴) +o (card‘𝐵))) | ||
| Theorem | ficardun 10241 | The cardinality of the union of disjoint, finite sets is the ordinal sum of their cardinalities. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Mario Carneiro, 28-Apr-2015.) Avoid ax-rep 5279. (Revised by BTernaryTau, 3-Jul-2024.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘(𝐴 ∪ 𝐵)) = ((card‘𝐴) +o (card‘𝐵))) | ||
| Theorem | ficardun2 10242 | The cardinality of the union of finite sets is at most the ordinal sum of their cardinalities. (Contributed by Mario Carneiro, 5-Feb-2013.) Avoid ax-rep 5279. (Revised by BTernaryTau, 3-Jul-2024.) |
| ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (card‘(𝐴 ∪ 𝐵)) ⊆ ((card‘𝐴) +o (card‘𝐵))) | ||
| Theorem | pwsdompw 10243* | Lemma for domtriom 10483. This is the equinumerosity version of the algebraic identity Σ𝑘 ∈ 𝑛(2↑𝑘) = (2↑𝑛) − 1. (Contributed by Mario Carneiro, 7-Feb-2013.) |
| ⊢ ((𝑛 ∈ ω ∧ ∀𝑘 ∈ suc 𝑛(𝐵‘𝑘) ≈ 𝒫 𝑘) → ∪ 𝑘 ∈ 𝑛 (𝐵‘𝑘) ≺ (𝐵‘𝑛)) | ||
| Theorem | unctb 10244 | The union of two countable sets is countable. (Contributed by FL, 25-Aug-2006.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) |
| ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 ∪ 𝐵) ≼ ω) | ||
| Theorem | infdjuabs 10245 | Absorption law for addition to an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≈ 𝐴) | ||
| Theorem | infunabs 10246 | An infinite set is equinumerous to its union with a smaller one. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴) | ||
| Theorem | infdju 10247 | The sum of two cardinal numbers is their maximum, if one of them is infinite. Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 ⊔ 𝐵) ≈ (𝐴 ∪ 𝐵)) | ||
| Theorem | infdif 10248 | The cardinality of an infinite set does not change after subtracting a strictly smaller one. Example in [Enderton] p. 164. (Contributed by NM, 22-Oct-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≺ 𝐴) → (𝐴 ∖ 𝐵) ≈ 𝐴) | ||
| Theorem | infdif2 10249 | Cardinality ordering for an infinite class difference. (Contributed by NM, 24-Mar-2007.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card ∧ ω ≼ 𝐴) → ((𝐴 ∖ 𝐵) ≼ 𝐵 ↔ 𝐴 ≼ 𝐵)) | ||
| Theorem | infxpdom 10250 | Dominance law for multiplication with an infinite cardinal. (Contributed by NM, 26-Mar-2006.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴) → (𝐴 × 𝐵) ≼ 𝐴) | ||
| Theorem | infxpabs 10251 | Absorption law for multiplication with an infinite cardinal. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴) | ||
| Theorem | infunsdom1 10252 | The union of two sets that are strictly dominated by the infinite set 𝑋 is also dominated by 𝑋. This version of infunsdom 10253 assumes additionally that 𝐴 is the smaller of the two. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
| ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋) | ||
| Theorem | infunsdom 10253 | The union of two sets that are strictly dominated by the infinite set 𝑋 is also strictly dominated by 𝑋. (Contributed by Mario Carneiro, 3-May-2015.) |
| ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴 ≺ 𝑋 ∧ 𝐵 ≺ 𝑋)) → (𝐴 ∪ 𝐵) ≺ 𝑋) | ||
| Theorem | infxp 10254 | Absorption law for multiplication with an infinite cardinal. Equivalent to Proposition 10.41 of [TakeutiZaring] p. 95. (Contributed by NM, 28-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
| ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ∈ dom card ∧ 𝐵 ≠ ∅)) → (𝐴 × 𝐵) ≈ (𝐴 ∪ 𝐵)) | ||
| Theorem | pwdjudom 10255 | A property of dominance over a powerset, and a main lemma for gchac 10721. Similar to Lemma 2.3 of [KanamoriPincus] p. 420. (Contributed by Mario Carneiro, 15-May-2015.) |
| ⊢ (𝒫 (𝐴 ⊔ 𝐴) ≼ (𝐴 ⊔ 𝐵) → 𝒫 𝐴 ≼ 𝐵) | ||
| Theorem | infpss 10256* | Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 10353. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
| ⊢ (ω ≼ 𝐴 → ∃𝑥(𝑥 ⊊ 𝐴 ∧ 𝑥 ≈ 𝐴)) | ||
| Theorem | infmap2 10257* | An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10616 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.) |
| ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) | ||
| Theorem | ackbij2lem1 10258 | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin)) | ||
| Theorem | ackbij1lem1 10259 | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ (¬ 𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = (𝐵 ∩ 𝐴)) | ||
| Theorem | ackbij1lem2 10260 | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ (𝐴 ∈ 𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵 ∩ 𝐴))) | ||
| Theorem | ackbij1lem3 10261 | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin)) | ||
| Theorem | ackbij1lem4 10262 | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
| ⊢ (𝐴 ∈ ω → {𝐴} ∈ (𝒫 ω ∩ Fin)) | ||
| Theorem | ackbij1lem5 10263 | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 19-Nov-2014.) (Proof shortened by AV, 18-Jul-2022.) |
| ⊢ (𝐴 ∈ ω → (card‘𝒫 suc 𝐴) = ((card‘𝒫 𝐴) +o (card‘𝒫 𝐴))) | ||
| Theorem | ackbij1lem6 10264 | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin)) | ||
| Theorem | ackbij1lem7 10265* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))) | ||
| Theorem | ackbij1lem8 10266* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴)) | ||
| Theorem | ackbij1lem9 10267* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 19-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +o (𝐹‘𝐵))) | ||
| Theorem | ackbij1lem10 10268* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω | ||
| Theorem | ackbij1lem11 10269* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (𝒫 ω ∩ Fin)) | ||
| Theorem | ackbij1lem12 10270* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)) | ||
| Theorem | ackbij1lem13 10271* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ (𝐹‘∅) = ∅ | ||
| Theorem | ackbij1lem14 10272* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴)) | ||
| Theorem | ackbij1lem15 10273* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝑐 ∈ ω ∧ 𝑐 ∈ 𝐴 ∧ ¬ 𝑐 ∈ 𝐵)) → ¬ (𝐹‘(𝐴 ∩ suc 𝑐)) = (𝐹‘(𝐵 ∩ suc 𝑐))) | ||
| Theorem | ackbij1lem16 10274* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵)) | ||
| Theorem | ackbij1lem17 10275* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω | ||
| Theorem | ackbij1lem18 10276* | Lemma for ackbij1 10277. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → ∃𝑏 ∈ (𝒫 ω ∩ Fin)(𝐹‘𝑏) = suc (𝐹‘𝐴)) | ||
| Theorem | ackbij1 10277* | The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1-onto→ω | ||
| Theorem | ackbij1b 10278* | The Ackermann bijection, part 1b: the bijection from ackbij1 10277 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) ⇒ ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)) | ||
| Theorem | ackbij2lem2 10279* | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) & ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦)))) ⇒ ⊢ (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴):(𝑅1‘𝐴)–1-1-onto→(card‘(𝑅1‘𝐴))) | ||
| Theorem | ackbij2lem3 10280* | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) & ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦)))) ⇒ ⊢ (𝐴 ∈ ω → (rec(𝐺, ∅)‘𝐴) ⊆ (rec(𝐺, ∅)‘suc 𝐴)) | ||
| Theorem | ackbij2lem4 10281* | Lemma for ackbij2 10282. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) & ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦)))) ⇒ ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (rec(𝐺, ∅)‘𝐵) ⊆ (rec(𝐺, ∅)‘𝐴)) | ||
| Theorem | ackbij2 10282* | The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦))) & ⊢ 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥 “ 𝑦)))) & ⊢ 𝐻 = ∪ (rec(𝐺, ∅) “ ω) ⇒ ⊢ 𝐻:∪ (𝑅1 “ ω)–1-1-onto→ω | ||
| Theorem | r1om 10283 | The set of hereditarily finite sets is countable. See ackbij2 10282 for an explicit bijection that works without Infinity. See also r1omALT 10816. (Contributed by Stefan O'Rear, 18-Nov-2014.) |
| ⊢ (𝑅1‘ω) ≈ ω | ||
| Theorem | fictb 10284 | A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.) |
| ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω)) | ||
| Theorem | cflem 10285* | A lemma used to simplify cofinality computations, showing the existence of the cardinal of an unbounded subset of a set 𝐴. (Contributed by NM, 24-Apr-2004.) Avoid ax-11 2157. (Revised by BTernaryTau, 25-Jul-2025.) |
| ⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))) | ||
| Theorem | cflemOLD 10286* | Obsolete version of cflem 10285 as of 25-Jul-2025. (Contributed by NM, 24-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ (𝐴 ∈ 𝑉 → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))) | ||
| Theorem | cfval 10287* | Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number 𝐴 is the cardinality (size) of the smallest unbounded subset 𝑦 of the ordinal number. Unbounded means that for every member of 𝐴, there is a member of 𝑦 that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) | ||
| Theorem | cff 10288 | Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.) |
| ⊢ cf:On⟶On | ||
| Theorem | cfub 10289* | An upper bound on cofinality. (Contributed by NM, 25-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} | ||
| Theorem | cflm 10290* | Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.) |
| ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))}) | ||
| Theorem | cf0 10291 | Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102. (Contributed by NM, 16-Apr-2004.) |
| ⊢ (cf‘∅) = ∅ | ||
| Theorem | cardcf 10292 | Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴) | ||
| Theorem | cflecard 10293 | Cofinality is bounded by the cardinality of its argument. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| ⊢ (cf‘𝐴) ⊆ (card‘𝐴) | ||
| Theorem | cfle 10294 | Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102. (Contributed by NM, 26-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.) |
| ⊢ (cf‘𝐴) ⊆ 𝐴 | ||
| Theorem | cfon 10295 | The cofinality of any set is an ordinal (although it only makes sense when 𝐴 is an ordinal). (Contributed by Mario Carneiro, 9-Mar-2013.) |
| ⊢ (cf‘𝐴) ∈ On | ||
| Theorem | cfeq0 10296 | Only the ordinal zero has cofinality zero. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.) |
| ⊢ (𝐴 ∈ On → ((cf‘𝐴) = ∅ ↔ 𝐴 = ∅)) | ||
| Theorem | cfsuc 10297 | Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102. (Contributed by NM, 23-Apr-2004.) (Revised by Mario Carneiro, 12-Feb-2013.) |
| ⊢ (𝐴 ∈ On → (cf‘suc 𝐴) = 1o) | ||
| Theorem | cff1 10298* | There is always a map from (cf‘𝐴) to 𝐴 (this is a stronger condition than the definition, which only presupposes a map from some 𝑦 ≈ (cf‘𝐴). (Contributed by Mario Carneiro, 28-Feb-2013.) |
| ⊢ (𝐴 ∈ On → ∃𝑓(𝑓:(cf‘𝐴)–1-1→𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ (cf‘𝐴)𝑧 ⊆ (𝑓‘𝑤))) | ||
| Theorem | cfflb 10299* | If there is a cofinal map from 𝐵 to 𝐴, then 𝐵 is at least (cf‘𝐴). This theorem and cff1 10298 motivate the picture of (cf‘𝐴) as the greatest lower bound of the domain of cofinal maps into 𝐴. (Contributed by Mario Carneiro, 28-Feb-2013.) |
| ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝑧 ⊆ (𝑓‘𝑤)) → (cf‘𝐴) ⊆ 𝐵)) | ||
| Theorem | cfval2 10300* | Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.) |
| ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤} (card‘𝑥)) | ||
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