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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cardennn 9401 | If 𝐴 is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.) |
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ∈ ω) → (card‘𝐴) = 𝐵) | ||
Theorem | cardsucinf 9402 | The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013.) |
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (card‘suc 𝐴) = (card‘𝐴)) | ||
Theorem | cardsucnn 9403 | The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9402. (Contributed by NM, 7-Nov-2008.) |
⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴)) | ||
Theorem | cardom 9404 | The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133. (Contributed by NM, 28-Oct-2003.) |
⊢ (card‘ω) = ω | ||
Theorem | carden2 9405 | Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 9962, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵)) | ||
Theorem | cardsdom2 9406 | A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴 ≺ 𝐵)) | ||
Theorem | domtri2 9407 | Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) | ||
Theorem | nnsdomel 9408 | Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ 𝐴 ≺ 𝐵)) | ||
Theorem | cardval2 9409* | An alternate version of the value of the cardinal number of a set. Compare cardval 9957. This theorem could be used to give a simpler definition of card in place of df-card 9357. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.) |
⊢ (𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥 ≺ 𝐴}) | ||
Theorem | isinffi 9410* | An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8720 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.) |
⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵–1-1→𝐴) | ||
Theorem | fidomtri 9411 | Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.) |
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) | ||
Theorem | fidomtri2 9412 | Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 7-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴)) | ||
Theorem | harsdom 9413 | The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴)) | ||
Theorem | onsdom 9414* | Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ dom card → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) | ||
Theorem | harval2 9415* | An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ dom card → (har‘𝐴) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) | ||
Theorem | cardmin2 9416* | The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.) |
⊢ (∃𝑥 ∈ On 𝐴 ≺ 𝑥 ↔ (card‘∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) = ∩ {𝑥 ∈ On ∣ 𝐴 ≺ 𝑥}) | ||
Theorem | pm54.43lem 9417* | In Theorem *54.43 of [WhiteheadRussell] p. 360, the number 1 is defined as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9386), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}. Here we show that this is equivalent to 𝐴 ≈ 1o so that we can use the latter more convenient notation in pm54.43 9418. (Contributed by NM, 4-Nov-2013.) |
⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) | ||
Theorem | pm54.43 9418 |
Theorem *54.43 of [WhiteheadRussell]
p. 360. "From this proposition it
will follow, when arithmetical addition has been defined, that
1+1=2."
See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations.
This theorem states that two sets of cardinality 1 are disjoint iff
their union has cardinality 2.
Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 9386), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} which is the same as 𝐴 ≈ 1o by pm54.43lem 9417. We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.) Theorem dju1p1e2 9588 shows the derivation of 1+1=2 for cardinal numbers from this theorem. (Contributed by NM, 4-Apr-2007.) |
⊢ ((𝐴 ≈ 1o ∧ 𝐵 ≈ 1o) → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐴 ∪ 𝐵) ≈ 2o)) | ||
Theorem | pr2nelem 9419 | Lemma for pr2ne 9420. (Contributed by FL, 17-Aug-2008.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | ||
Theorem | pr2ne 9420 | If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | prdom2 9421 | An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o) | ||
Theorem | en2eqpr 9422 | Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
⊢ ((𝐶 ≈ 2o ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ≠ 𝐵 → 𝐶 = {𝐴, 𝐵})) | ||
Theorem | en2eleq 9423 | Express a set of pair cardinality as the unordered pair of a given element and the other element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})}) | ||
Theorem | en2other2 9424 | Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2o) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋) | ||
Theorem | dif1card 9425 | The cardinality of a nonempty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.) |
⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))) | ||
Theorem | leweon 9426* | Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9427, this order is not set-like, as the preimage of 〈1o, ∅〉 is the proper class ({∅} × On). (Contributed by Mario Carneiro, 9-Mar-2013.) |
⊢ 𝐿 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))} ⇒ ⊢ 𝐿 We (On × On) | ||
Theorem | r0weon 9427* | A set-like well-ordering of the class of ordinal pairs. Proposition 7.58(1) of [TakeutiZaring] p. 54. (Contributed by Mario Carneiro, 7-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐿 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))} & ⊢ 𝑅 = {〈𝑧, 𝑤〉 ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∨ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))} ⇒ ⊢ (𝑅 We (On × On) ∧ 𝑅 Se (On × On)) | ||
Theorem | infxpenlem 9428* | Lemma for infxpen 9429. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ 𝐿 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1st ‘𝑥) ∈ (1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥) ∈ (2nd ‘𝑦))))} & ⊢ 𝑅 = {〈𝑧, 𝑤〉 ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) ∈ ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∨ (((1st ‘𝑧) ∪ (2nd ‘𝑧)) = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∧ 𝑧𝐿𝑤)))} & ⊢ 𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎))) & ⊢ (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚 ∈ 𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚 ∈ 𝑎 𝑚 ≺ 𝑎))) & ⊢ 𝑀 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) & ⊢ 𝐽 = OrdIso(𝑄, (𝑎 × 𝑎)) ⇒ ⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) | ||
Theorem | infxpen 9429 | Every infinite ordinal is equinumerous to its Cartesian square. Proposition 10.39 of [TakeutiZaring] p. 94, whose proof we follow closely. The key idea is to show that the relation 𝑅 is a well-ordering of (On × On) with the additional property that 𝑅-initial segments of (𝑥 × 𝑥) (where 𝑥 is a limit ordinal) are of cardinality at most 𝑥. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Jun-2015.) |
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) | ||
Theorem | xpomen 9430 | The Cartesian product of omega (the set of ordinal natural numbers) with itself is equinumerous to omega. Exercise 1 of [Enderton] p. 133. (Contributed by NM, 23-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.) |
⊢ (ω × ω) ≈ ω | ||
Theorem | xpct 9431 | The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.) |
⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω) | ||
Theorem | infxpidm2 9432 | The Cartesian product of an infinite set with itself is idempotent. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 9973. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) | ||
Theorem | infxpenc 9433* | A canonical version of infxpen 9429, by a completely different approach (although it uses infxpen 9429 via xpomen 9430). Using Cantor's normal form, we can show that 𝐴 ↑o 𝐵 respects equinumerosity (oef1o 9150), so that all the steps of (ω↑𝑊) · (ω↑𝑊) ≈ ω↑(2𝑊) ≈ (ω↑2)↑𝑊 ≈ ω↑𝑊 can be verified using bijections to do the ordinal commutations. (The assumption on 𝑁 can be satisfied using cnfcom3c 9158.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ω ⊆ 𝐴) & ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o)) & ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω) & ⊢ (𝜑 → (𝐹‘∅) = ∅) & ⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑o 𝑊)) & ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊)))) & ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊)) & ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋)))) & ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤)) & ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) & ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o))) & ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦)) & ⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ 〈(𝑁‘𝑥), (𝑁‘𝑦)〉) & ⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)) ⇒ ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴) | ||
Theorem | infxpenc2lem1 9434* | Lemma for infxpenc2 9437. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) & ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏)) ⇒ ⊢ ((𝜑 ∧ (𝑏 ∈ 𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑊))) | ||
Theorem | infxpenc2lem2 9435* | Lemma for infxpenc2 9437. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) & ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏)) & ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω) & ⊢ (𝜑 → (𝐹‘∅) = ∅) & ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊)))) & ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊)) & ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋)))) & ⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤)) & ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧)) & ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o))) & ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦)) & ⊢ 𝑇 = (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ 〈((𝑛‘𝑏)‘𝑥), ((𝑛‘𝑏)‘𝑦)〉) & ⊢ 𝐺 = (◡(𝑛‘𝑏) ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)) ⇒ ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) | ||
Theorem | infxpenc2lem3 9436* | Lemma for infxpenc2 9437. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.) |
⊢ (𝜑 → 𝐴 ∈ On) & ⊢ (𝜑 → ∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤))) & ⊢ 𝑊 = (◡(𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛‘𝑏)) & ⊢ (𝜑 → 𝐹:(ω ↑o 2o)–1-1-onto→ω) & ⊢ (𝜑 → (𝐹‘∅) = ∅) ⇒ ⊢ (𝜑 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) | ||
Theorem | infxpenc2 9437* | Existence form of infxpenc 9433. A "uniform" or "canonical" version of infxpen 9429, asserting the existence of a single function 𝑔 that simultaneously demonstrates product idempotence of all ordinals below a given bound. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → (𝑔‘𝑏):(𝑏 × 𝑏)–1-1-onto→𝑏)) | ||
Theorem | iunmapdisj 9438* | The union ∪ 𝑛 ∈ 𝐶(𝐴 ↑m 𝑛) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.) |
⊢ ∃*𝑛 ∈ 𝐶 𝐵 ∈ (𝐴 ↑m 𝑛) | ||
Theorem | fseqenlem1 9439* | Lemma for fseqen 9442. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴) & ⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {〈∅, 𝐵〉}) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ ω) → (𝐺‘𝐶):(𝐴 ↑m 𝐶)–1-1→𝐴) | ||
Theorem | fseqenlem2 9440* | Lemma for fseqen 9442. (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴) & ⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {〈∅, 𝐵〉}) & ⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↦ 〈dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)〉) ⇒ ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)–1-1→(ω × 𝐴)) | ||
Theorem | fseqdom 9441* | One half of fseqen 9442. (Contributed by Mario Carneiro, 18-Nov-2014.) |
⊢ (𝐴 ∈ 𝑉 → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) | ||
Theorem | fseqen 9442* | A set that is equinumerous to its Cartesian product is equinumerous to the set of finite sequences on it. (This can be proven more easily using some choice but this proof avoids it.) (Contributed by Mario Carneiro, 18-Nov-2014.) |
⊢ (((𝐴 × 𝐴) ≈ 𝐴 ∧ 𝐴 ≠ ∅) → ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ≈ (ω × 𝐴)) | ||
Theorem | infpwfidom 9443 | The collection of finite subsets of a set dominates the set. (We use the weaker sethood assumption (𝒫 𝐴 ∩ Fin) ∈ V because this theorem also implies that 𝐴 is a set if 𝒫 𝐴 ∩ Fin is.) (Contributed by Mario Carneiro, 17-May-2015.) |
⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → 𝐴 ≼ (𝒫 𝐴 ∩ Fin)) | ||
Theorem | dfac8alem 9444* | Lemma for dfac8a 9445. If the power set of a set has a choice function, then the set is numerable. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.) |
⊢ 𝐹 = recs(𝐺) & ⊢ 𝐺 = (𝑓 ∈ V ↦ (𝑔‘(𝐴 ∖ ran 𝑓))) ⇒ ⊢ (𝐴 ∈ 𝐶 → (∃𝑔∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card)) | ||
Theorem | dfac8a 9445* | Numeration theorem: every set with a choice function on its power set is numerable. With AC, this reduces to the statement that every set is numerable. Similar to Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 5-Jan-2013.) |
⊢ (𝐴 ∈ 𝐵 → (∃ℎ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (ℎ‘𝑦) ∈ 𝑦) → 𝐴 ∈ dom card)) | ||
Theorem | dfac8b 9446* | The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴) | ||
Theorem | dfac8clem 9447* | Lemma for dfac8c 9448. (Contributed by Mario Carneiro, 10-Jan-2013.) |
⊢ 𝐹 = (𝑠 ∈ (𝐴 ∖ {∅}) ↦ (℩𝑎 ∈ 𝑠 ∀𝑏 ∈ 𝑠 ¬ 𝑏𝑟𝑎)) ⇒ ⊢ (𝐴 ∈ 𝐵 → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) | ||
Theorem | dfac8c 9448* | If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.) |
⊢ (𝐴 ∈ 𝐵 → (∃𝑟 𝑟 We ∪ 𝐴 → ∃𝑓∀𝑧 ∈ 𝐴 (𝑧 ≠ ∅ → (𝑓‘𝑧) ∈ 𝑧))) | ||
Theorem | ac10ct 9449* | A proof of the well-ordering theorem weth 9906, an Axiom of Choice equivalent, restricted to sets dominated by some ordinal (in particular finite sets and countable sets), proven in ZF without AC. (Contributed by Mario Carneiro, 5-Jan-2013.) |
⊢ (∃𝑦 ∈ On 𝐴 ≼ 𝑦 → ∃𝑥 𝑥 We 𝐴) | ||
Theorem | ween 9450* | A set is numerable iff it can be well-ordered. (Contributed by Mario Carneiro, 5-Jan-2013.) |
⊢ (𝐴 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐴) | ||
Theorem | ac5num 9451* | A version of ac5b 9889 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ ((∪ 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴⟶∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝑥)) | ||
Theorem | ondomen 9452 | If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) | ||
Theorem | numdom 9453 | A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) | ||
Theorem | ssnum 9454 | A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ dom card) | ||
Theorem | onssnum 9455 | All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card) | ||
Theorem | indcardi 9456* | Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ dom card) & ⊢ ((𝜑 ∧ 𝑅 ≼ 𝑇 ∧ ∀𝑦(𝑆 ≺ 𝑅 → 𝜒)) → 𝜓) & ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) & ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) & ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆) & ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇) ⇒ ⊢ (𝜑 → 𝜃) | ||
Theorem | acnrcl 9457 | Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V) | ||
Theorem | acneq 9458 | Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶) | ||
Theorem | isacn 9459* | The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))) | ||
Theorem | acni 9460* | The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝑋 ∈ AC 𝐴 ∧ 𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝐹‘𝑥)) | ||
Theorem | acni2 9461* | The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐵 ⊆ 𝑋 ∧ 𝐵 ≠ ∅)) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ 𝐵)) | ||
Theorem | acni3 9462* | The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑦 = (𝑔‘𝑥) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑋 ∈ AC 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑋 𝜑) → ∃𝑔(𝑔:𝐴⟶𝑋 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | ||
Theorem | acnlem 9463* | Construct a mapping satisfying the consequent of isacn 9459. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) | ||
Theorem | numacn 9464 | A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 ∈ 𝑉 → (𝑋 ∈ dom card → 𝑋 ∈ AC 𝐴)) | ||
Theorem | finacn 9465 | Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 ∈ Fin → AC 𝐴 = V) | ||
Theorem | acndom 9466 | A set with long choice sequences also has shorter choice sequences, where "shorter" here means the new index set is dominated by the old index set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 ≼ 𝐵 → (𝑋 ∈ AC 𝐵 → 𝑋 ∈ AC 𝐴)) | ||
Theorem | acnnum 9467 | A set 𝑋 which has choice sequences on it of length 𝒫 𝑋 is well-orderable (and hence has choice sequences of every length). (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑋 ∈ AC 𝒫 𝑋 ↔ 𝑋 ∈ dom card) | ||
Theorem | acnen 9468 | The class of choice sets of length 𝐴 is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 ≈ 𝐵 → AC 𝐴 = AC 𝐵) | ||
Theorem | acndom2 9469 | A set smaller than one with choice sequences of length 𝐴 also has choice sequences of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑋 ≼ 𝑌 → (𝑌 ∈ AC 𝐴 → 𝑋 ∈ AC 𝐴)) | ||
Theorem | acnen2 9470 | The class of sets with choice sequences of length 𝐴 is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑋 ≈ 𝑌 → (𝑋 ∈ AC 𝐴 ↔ 𝑌 ∈ AC 𝐴)) | ||
Theorem | fodomacn 9471 | A version of fodom 9933 that doesn't require the Axiom of Choice ax-ac 9870. If 𝐴 has choice sequences of length 𝐵, then any surjection from 𝐴 to 𝐵 can be inverted to an injection the other way. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 ∈ AC 𝐵 → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | ||
Theorem | fodomnum 9472 | A version of fodom 9933 that doesn't require the Axiom of Choice ax-ac 9870. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.) |
⊢ (𝐴 ∈ dom card → (𝐹:𝐴–onto→𝐵 → 𝐵 ≼ 𝐴)) | ||
Theorem | fonum 9473 | A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ∈ dom card) | ||
Theorem | numwdom 9474 | A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ≼* 𝐴) → 𝐵 ∈ dom card) | ||
Theorem | fodomfi2 9475 | Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ Fin ∧ 𝐹:𝐴–onto→𝐵) → 𝐵 ≼ 𝐴) | ||
Theorem | wdomfil 9476 | Weak dominance agrees with normal for finite left sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝑋 ∈ Fin → (𝑋 ≼* 𝑌 ↔ 𝑋 ≼ 𝑌)) | ||
Theorem | infpwfien 9477 | Any infinite well-orderable set is equinumerous to its set of finite subsets. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝒫 𝐴 ∩ Fin) ≈ 𝐴) | ||
Theorem | inffien 9478 | The set of finite intersections of an infinite well-orderable set is equinumerous to the set itself. (Contributed by Mario Carneiro, 18-May-2015.) |
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (fi‘𝐴) ≈ 𝐴) | ||
Theorem | wdomnumr 9479 | Weak dominance agrees with normal for numerable right sets. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.) |
⊢ (𝐵 ∈ dom card → (𝐴 ≼* 𝐵 ↔ 𝐴 ≼ 𝐵)) | ||
Theorem | alephfnon 9480 | The aleph function is a function on the class of ordinal numbers. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ ℵ Fn On | ||
Theorem | aleph0 9481 | The first infinite cardinal number, discovered by Georg Cantor in 1873, has the same size as the set of natural numbers ω (and under our particular definition is also equal to it). In the literature, the argument of the aleph function is often written as a subscript, and the first aleph is written ℵ0. Exercise 3 of [TakeutiZaring] p. 91. Also Definition 12(i) of [Suppes] p. 228. From Moshé Machover, Set Theory, Logic, and Their Limitations, p. 95: "Aleph...the first letter in the Hebrew alphabet...is also the first letter of the Hebrew word...(einsoph, meaning infinity), which is a cabbalistic appellation of the deity. The notation is due to Cantor, who was deeply interested in mysticism." (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ (ℵ‘∅) = ω | ||
Theorem | alephlim 9482* | Value of the aleph function at a limit ordinal. Definition 12(iii) of [Suppes] p. 91. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑥 ∈ 𝐴 (ℵ‘𝑥)) | ||
Theorem | alephsuc 9483 | Value of the aleph function at a successor ordinal. Definition 12(ii) of [Suppes] p. 91. Here we express the successor aleph in terms of the Hartogs function df-har 9011, which gives the smallest ordinal that strictly dominates its argument (or the supremum of all ordinals that are dominated by the argument). (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = (har‘(ℵ‘𝐴))) | ||
Theorem | alephon 9484 | An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
⊢ (ℵ‘𝐴) ∈ On | ||
Theorem | alephcard 9485 | Every aleph is a cardinal number. Theorem 65 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
⊢ (card‘(ℵ‘𝐴)) = (ℵ‘𝐴) | ||
Theorem | alephnbtwn 9486 | No cardinal can be sandwiched between an aleph and its successor aleph. Theorem 67 of [Suppes] p. 229. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ ((card‘𝐵) = 𝐵 → ¬ ((ℵ‘𝐴) ∈ 𝐵 ∧ 𝐵 ∈ (ℵ‘suc 𝐴))) | ||
Theorem | alephnbtwn2 9487 | No set has equinumerosity between an aleph and its successor aleph. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
⊢ ¬ ((ℵ‘𝐴) ≺ 𝐵 ∧ 𝐵 ≺ (ℵ‘suc 𝐴)) | ||
Theorem | alephordilem1 9488 | Lemma for alephordi 9489. (Contributed by NM, 23-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) | ||
Theorem | alephordi 9489 | Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵))) | ||
Theorem | alephord 9490 | Ordering property of the aleph function. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 9-Feb-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵))) | ||
Theorem | alephord2 9491 | Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ (ℵ‘𝐴) ∈ (ℵ‘𝐵))) | ||
Theorem | alephord2i 9492 | Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.) |
⊢ (𝐵 ∈ On → (𝐴 ∈ 𝐵 → (ℵ‘𝐴) ∈ (ℵ‘𝐵))) | ||
Theorem | alephord3 9493 | Ordering property of the aleph function. (Contributed by NM, 11-Nov-2003.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (ℵ‘𝐴) ⊆ (ℵ‘𝐵))) | ||
Theorem | alephsucdom 9494 | A set dominated by an aleph is strictly dominated by its successor aleph and vice-versa. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
⊢ (𝐵 ∈ On → (𝐴 ≼ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘suc 𝐵))) | ||
Theorem | alephsuc2 9495* | An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 8997 function by transfinite recursion, starting from ω. Using this theorem we could define the aleph function with {𝑧 ∈ On ∣ 𝑧 ≼ 𝑥} in place of ∩ {𝑧 ∈ On ∣ 𝑥 ≺ 𝑧} in df-aleph 9358. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.) |
⊢ (𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)}) | ||
Theorem | alephdom 9496 | Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (ℵ‘𝐴) ≼ (ℵ‘𝐵))) | ||
Theorem | alephgeom 9497 | Every aleph is greater than or equal to the set of natural numbers. (Contributed by NM, 11-Nov-2003.) |
⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴)) | ||
Theorem | alephislim 9498 | Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.) |
⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴)) | ||
Theorem | aleph11 9499 | The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) = (ℵ‘𝐵) ↔ 𝐴 = 𝐵)) | ||
Theorem | alephf1 9500 | The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 9518. (Contributed by Mario Carneiro, 2-Feb-2013.) |
⊢ ℵ:On–1-1→On |
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