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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cardnueq0 10001 | The empty set is the only numerable set with cardinality zero. (Contributed by Mario Carneiro, 7-Jan-2013.) |
⊢ (𝐴 ∈ dom card → ((card‘𝐴) = ∅ ↔ 𝐴 = ∅)) | ||
Theorem | cardne 10002 | No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.) |
⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵) | ||
Theorem | carden2a 10003 | If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 10004 are meant to replace carden 10588 in ZF without AC. (Contributed by Mario Carneiro, 9-Jan-2013.) |
⊢ (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴 ≈ 𝐵) | ||
Theorem | carden2b 10004 | If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 10003 are meant to replace carden 10588 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) |
⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵)) | ||
Theorem | card1 10005* | A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.) |
⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | cardsn 10006 | A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.) |
⊢ (𝐴 ∈ 𝑉 → (card‘{𝐴}) = 1o) | ||
Theorem | carddomi2 10007 | Two sets have the dominance relationship if their cardinalities have the subset relationship and one is numerable. See also carddom 10591, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ 𝑉) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵)) | ||
Theorem | sdomsdomcardi 10008 | A set strictly dominates if its cardinal strictly dominates. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ (𝐴 ≺ (card‘𝐵) → 𝐴 ≺ 𝐵) | ||
Theorem | cardlim 10009 | An infinite cardinal is a limit ordinal. Equivalent to Exercise 4 of [TakeutiZaring] p. 91. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ (ω ⊆ (card‘𝐴) ↔ Lim (card‘𝐴)) | ||
Theorem | cardsdomelir 10010 | A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 10011 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵) | ||
Theorem | cardsdomel 10011 | A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 ↔ 𝐴 ∈ (card‘𝐵))) | ||
Theorem | iscard 10012* | Two ways to express the property of being a cardinal number. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ 𝐴 𝑥 ≺ 𝐴)) | ||
Theorem | iscard2 10013* | Two ways to express the property of being a cardinal number. Definition 8 of [Suppes] p. 225. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ ((card‘𝐴) = 𝐴 ↔ (𝐴 ∈ On ∧ ∀𝑥 ∈ On (𝐴 ≈ 𝑥 → 𝐴 ⊆ 𝑥))) | ||
Theorem | carddom2 10014 | Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10591, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵)) | ||
Theorem | harcard 10015 | The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ (card‘(har‘𝐴)) = (har‘𝐴) | ||
Theorem | cardprclem 10016* | Lemma for cardprc 10017. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.) |
⊢ 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥} ⇒ ⊢ ¬ 𝐴 ∈ V | ||
Theorem | cardprc 10017 | The class of all cardinal numbers is not a set (i.e. is a proper class). Theorem 19.8 of [Eisenberg] p. 310. In this proof (which does not use AC), we cannot use Cantor's construction canth3 10598 to ensure that there is always a cardinal larger than a given cardinal, but we can use Hartogs' construction hartogs 9581 to construct (effectively) (ℵ‘suc 𝐴) from (ℵ‘𝐴), which achieves the same thing. (Contributed by Mario Carneiro, 22-Jan-2013.) |
⊢ {𝑥 ∣ (card‘𝑥) = 𝑥} ∉ V | ||
Theorem | carduni 10018* | The union of a set of cardinals is a cardinal. Theorem 18.14 of [Monk1] p. 133. (Contributed by Mario Carneiro, 20-Jan-2013.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝑥) = 𝑥 → (card‘∪ 𝐴) = ∪ 𝐴)) | ||
Theorem | cardiun 10019* | The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.) |
⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)) | ||
Theorem | cardennn 10020 | 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 10021 | 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 10022 | The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 10021. (Contributed by NM, 7-Nov-2008.) |
⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴)) | ||
Theorem | cardom 10023 | 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 10024 | Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 10588, 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 10025 | 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 10026 | 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 10027 | 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 10028* | An alternate version of the value of the cardinal number of a set. Compare cardval 10583. This theorem could be used to give a simpler definition of card in place of df-card 9976. 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 10029* | An infinite set contains subsets equinumerous to every finite set. Extension of isinf 9293 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 10030 | 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 10031 | 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 10032 | 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 10033* | 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 10034* | 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 | harsucnn 10035 | The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.) |
⊢ (𝐴 ∈ ω → (har‘𝐴) = suc 𝐴) | ||
Theorem | cardmin2 10036* | 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 10037* | 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 10005), 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 10038. (Contributed by NM, 4-Nov-2013.) |
⊢ (𝐴 ≈ 1o ↔ 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o}) | ||
Theorem | pm54.43 10038 |
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 10005), so that their 𝐴 ∈ 1 means, in our notation, 𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o} which is the same as 𝐴 ≈ 1o by pm54.43lem 10037. 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 10211 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 | enpr2 10039 | An unordered pair with distinct elements is equinumerous to ordinal two. This is a closed-form version of enpr2d 9087. (Contributed by FL, 17-Aug-2008.) Avoid ax-pow 5370, ax-un 7753. (Revised by BTernaryTau, 30-Dec-2024.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | ||
Theorem | pr2nelemOLD 10040 | Obsolete version of enpr2 10039 as of 30-Dec-2024. (Contributed by FL, 17-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴 ≠ 𝐵) → {𝐴, 𝐵} ≈ 2o) | ||
Theorem | pr2ne 10041 | If an unordered pair has two elements, then they are different. (Contributed by FL, 14-Feb-2010.) Avoid ax-pow 5370, ax-un 7753. (Revised by BTernaryTau, 30-Dec-2024.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | pr2neOLD 10042 | Obsolete version of pr2ne 10041 as of 30-Dec-2024. (Contributed by FL, 14-Feb-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | ||
Theorem | prdom2 10043 | An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o) | ||
Theorem | en2eqpr 10044 | 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 10045 | 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 10046 | 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 10047 | 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 10048* | Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 10049, 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 10049* | 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 10050* | Lemma for infxpen 10051. (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 10051 | 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 10052 | 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 10053 | 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 10054 | Every infinite well-orderable set is equinumerous to its Cartesian square. This theorem provides the basis for infinite cardinal arithmetic. Proposition 10.40 of [TakeutiZaring] p. 95. See also infxpidm 10599. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
⊢ ((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴) | ||
Theorem | infxpenc 10055* | A canonical version of infxpen 10051, by a completely different approach (although it uses infxpen 10051 via xpomen 10052). Using Cantor's normal form, we can show that 𝐴 ↑o 𝐵 respects equinumerosity (oef1o 9735), 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 9743.) (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 10056* | Lemma for infxpenc2 10059. (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 10057* | Lemma for infxpenc2 10059. (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 10058* | Lemma for infxpenc2 10059. (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 10059* | Existence form of infxpenc 10055. A "uniform" or "canonical" version of infxpen 10051, 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 10060* | 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 10061* | Lemma for fseqen 10064. (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 10062* | Lemma for fseqen 10064. (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 10063* | One half of fseqen 10064. (Contributed by Mario Carneiro, 18-Nov-2014.) |
⊢ (𝐴 ∈ 𝑉 → (ω × 𝐴) ≼ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛)) | ||
Theorem | fseqen 10064* | 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 10065 | 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 10066* | Lemma for dfac8a 10067. 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 10067* | 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 10068* | 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 10069* | Lemma for dfac8c 10070. (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 10070* | 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 10071* | A proof of the well-ordering theorem weth 10532, 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 10072* | 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 10073* | A version of ac5b 10515 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 10074 | 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 10075 | 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 10076 | 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 10077 | All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ⊆ On) → 𝐴 ∈ dom card) | ||
Theorem | indcardi 10078* | 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 10079 | Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑋 ∈ AC 𝐴 → 𝐴 ∈ V) | ||
Theorem | acneq 10080 | Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 𝐶 → AC 𝐴 = AC 𝐶) | ||
Theorem | isacn 10081* | 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 10082* | 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 10083* | 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 10084* | 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 10085* | Construct a mapping satisfying the consequent of isacn 10081. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (𝑓‘𝑥)) → ∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) | ||
Theorem | numacn 10086 | 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 10087 | Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 ∈ Fin → AC 𝐴 = V) | ||
Theorem | acndom 10088 | 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 10089 | 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 10090 | 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 10091 | 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 10092 | 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 10093 | A version of fodom 10560 that doesn't require the Axiom of Choice ax-ac 10496. 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 10094 | A version of fodom 10560 that doesn't require the Axiom of Choice ax-ac 10496. (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 10095 | 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 10096 | 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 10097 | 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 10098 | 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 10099 | 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 10100 | 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) |
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