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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcardsucinf 9401 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))

Theoremcardsucnn 9402 The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9401. (Contributed by NM, 7-Nov-2008.)
(𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))

Theoremcardom 9403 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

Theoremcarden2 9404 Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 9966, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))

Theoremcardsdom2 9405 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) ↔ 𝐴𝐵))

Theoremdomtri2 9406 Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremnnsdomel 9407 Strict dominance and elementhood are the same for finite ordinals. (Contributed by Stefan O'Rear, 2-Nov-2014.)
((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴𝐵))

Theoremcardval2 9408* An alternate version of the value of the cardinal number of a set. Compare cardval 9961. This theorem could be used to give a simpler definition of card in place of df-card 9356. It apparently does not occur in the literature. (Contributed by NM, 7-Nov-2003.)
(𝐴 ∈ dom card → (card‘𝐴) = {𝑥 ∈ On ∣ 𝑥𝐴})

Theoremisinffi 9409* An infinite set contains subsets equinumerous to every finite set. Extension of isinf 8719 from finite ordinals to all finite sets. (Contributed by Stefan O'Rear, 8-Oct-2014.)
((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐵1-1𝐴)

Theoremfidomtri 9410 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 ∧ 𝐵𝑉) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremfidomtri2 9411 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.)
((𝐴𝑉𝐵 ∈ Fin) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Theoremharsdom 9412 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))

Theoremonsdom 9413* 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 𝐴𝑥)

Theoremharval2 9414* 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 ∣ 𝐴𝑥})

Theoremharsucnn 9415 The next cardinal after a finite ordinal is the successor ordinal. (Contributed by RP, 5-Nov-2023.)
(𝐴 ∈ ω → (har‘𝐴) = suc 𝐴)

Theoremcardmin2 9416* The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)
(∃𝑥 ∈ On 𝐴𝑥 ↔ (card‘ {𝑥 ∈ On ∣ 𝐴𝑥}) = {𝑥 ∈ On ∣ 𝐴𝑥})

Theorempm54.43lem 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 9385), 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 ≈ 1o𝐴 ∈ {𝑥 ∣ (card‘𝑥) = 1o})

Theorempm54.43 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 9385), 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 ≈ 1o𝐵 ≈ 1o) → ((𝐴𝐵) = ∅ ↔ (𝐴𝐵) ≈ 2o))

Theorempr2nelem 9419 Lemma for pr2ne 9420. (Contributed by FL, 17-Aug-2008.)
((𝐴𝐶𝐵𝐷𝐴𝐵) → {𝐴, 𝐵} ≈ 2o)

Theorempr2ne 9420 If an unordered pair has two elements they are different. (Contributed by FL, 14-Feb-2010.)
((𝐴𝐶𝐵𝐷) → ({𝐴, 𝐵} ≈ 2o𝐴𝐵))

Theoremprdom2 9421 An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.)
((𝐴𝐶𝐵𝐷) → {𝐴, 𝐵} ≼ 2o)

Theoremen2eqpr 9422 Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
((𝐶 ≈ 2o𝐴𝐶𝐵𝐶) → (𝐴𝐵𝐶 = {𝐴, 𝐵}))

Theoremen2eleq 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.)
((𝑋𝑃𝑃 ≈ 2o) → 𝑃 = {𝑋, (𝑃 ∖ {𝑋})})

Theoremen2other2 9424 Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝑋𝑃𝑃 ≈ 2o) → (𝑃 ∖ { (𝑃 ∖ {𝑋})}) = 𝑋)

Theoremdif1card 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 ∧ 𝑋𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))

Theoremleweon 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)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}       𝐿 We (On × On)

Theoremr0weon 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)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}    &   𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}       (𝑅 We (On × On) ∧ 𝑅 Se (On × On))

Theoreminfxpenlem 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)) ∧ ((1st𝑥) ∈ (1st𝑦) ∨ ((1st𝑥) = (1st𝑦) ∧ (2nd𝑥) ∈ (2nd𝑦))))}    &   𝑅 = {⟨𝑧, 𝑤⟩ ∣ ((𝑧 ∈ (On × On) ∧ 𝑤 ∈ (On × On)) ∧ (((1st𝑧) ∪ (2nd𝑧)) ∈ ((1st𝑤) ∪ (2nd𝑤)) ∨ (((1st𝑧) ∪ (2nd𝑧)) = ((1st𝑤) ∪ (2nd𝑤)) ∧ 𝑧𝐿𝑤)))}    &   𝑄 = (𝑅 ∩ ((𝑎 × 𝑎) × (𝑎 × 𝑎)))    &   (𝜑 ↔ ((𝑎 ∈ On ∧ ∀𝑚𝑎 (ω ⊆ 𝑚 → (𝑚 × 𝑚) ≈ 𝑚)) ∧ (ω ⊆ 𝑎 ∧ ∀𝑚𝑎 𝑚𝑎)))    &   𝑀 = ((1st𝑤) ∪ (2nd𝑤))    &   𝐽 = OrdIso(𝑄, (𝑎 × 𝑎))       ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Theoreminfxpen 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)

Theoremxpomen 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

Theoremxpct 9431 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝐴 × 𝐵) ≼ ω)

Theoreminfxpidm2 9432 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 9977. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → (𝐴 × 𝐴) ≈ 𝐴)

Theoreminfxpenc 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 𝐴o 𝐵 respects equinumerosity (oef1o 9149), 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 9157.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ω ⊆ 𝐴)    &   (𝜑𝑊 ∈ (On ∖ 1o))    &   (𝜑𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   (𝜑 → (𝐹‘∅) = ∅)    &   (𝜑𝑁:𝐴1-1-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(𝑌𝑋))))    &   𝑋 = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))    &   𝑌 = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((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 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)    &   𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))       (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)

Theoreminfxpenc2lem1 9434* Lemma for infxpenc2 9437. (Contributed by Mario Carneiro, 30-May-2015.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))    &   𝑊 = ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛𝑏))       ((𝜑 ∧ (𝑏𝐴 ∧ ω ⊆ 𝑏)) → (𝑊 ∈ (On ∖ 1o) ∧ (𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑊)))

Theoreminfxpenc2lem2 9435* Lemma for infxpenc2 9437. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))    &   𝑊 = ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛𝑏))    &   (𝜑𝐹:(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(𝑌𝑋))))    &   𝑋 = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))    &   𝑌 = (𝑧 ∈ 2o, 𝑤𝑊 ↦ ((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 = (𝑥𝑏, 𝑦𝑏 ↦ ⟨((𝑛𝑏)‘𝑥), ((𝑛𝑏)‘𝑦)⟩)    &   𝐺 = ((𝑛𝑏) ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))       (𝜑 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))

Theoreminfxpenc2lem3 9436* Lemma for infxpenc2 9437. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)
(𝜑𝐴 ∈ On)    &   (𝜑 → ∀𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑛𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))    &   𝑊 = ((𝑥 ∈ (On ∖ 1o) ↦ (ω ↑o 𝑥))‘ran (𝑛𝑏))    &   (𝜑𝐹:(ω ↑o 2o)–1-1-onto→ω)    &   (𝜑 → (𝐹‘∅) = ∅)       (𝜑 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))

Theoreminfxpenc2 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 → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → (𝑔𝑏):(𝑏 × 𝑏)–1-1-onto𝑏))

Theoremiunmapdisj 9438* The union 𝑛𝐶(𝐴m 𝑛) is a disjoint union. (Contributed by Mario Carneiro, 17-May-2015.) (Revised by NM, 16-Jun-2017.)
∃*𝑛𝐶 𝐵 ∈ (𝐴m 𝑛)

Theoremfseqenlem1 9439* Lemma for fseqen 9442. (Contributed by Mario Carneiro, 17-May-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝐴)    &   (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)    &   𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})       ((𝜑𝐶 ∈ ω) → (𝐺𝐶):(𝐴m 𝐶)–1-1𝐴)

Theoremfseqenlem2 9440* Lemma for fseqen 9442. (Contributed by Mario Carneiro, 17-May-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝐴)    &   (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)    &   𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})    &   𝐾 = (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)       (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴))

Theoremfseqdom 9441* One half of fseqen 9442. (Contributed by Mario Carneiro, 18-Nov-2014.)
(𝐴𝑉 → (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴m 𝑛))

Theoremfseqen 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) ≈ 𝐴𝐴 ≠ ∅) → 𝑛 ∈ ω (𝐴m 𝑛) ≈ (ω × 𝐴))

Theoreminfpwfidom 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))

Theoremdfac8alem 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)))       (𝐴𝐶 → (∃𝑔𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑔𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Theoremdfac8a 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.)
(𝐴𝐵 → (∃𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝑦) ∈ 𝑦) → 𝐴 ∈ dom card))

Theoremdfac8b 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)

Theoremdfac8clem 9447* Lemma for dfac8c 9448. (Contributed by Mario Carneiro, 10-Jan-2013.)
𝐹 = (𝑠 ∈ (𝐴 ∖ {∅}) ↦ (𝑎𝑠𝑏𝑠 ¬ 𝑏𝑟𝑎))       (𝐴𝐵 → (∃𝑟 𝑟 We 𝐴 → ∃𝑓𝑧𝐴 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))

Theoremdfac8c 9448* If the union of a set is well-orderable, then the set has a choice function. (Contributed by Mario Carneiro, 5-Jan-2013.)
(𝐴𝐵 → (∃𝑟 𝑟 We 𝐴 → ∃𝑓𝑧𝐴 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))

Theoremac10ct 9449* A proof of the well-ordering theorem weth 9910, 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 𝐴𝑦 → ∃𝑥 𝑥 We 𝐴)

Theoremween 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)

Theoremac5num 9451* A version of ac5b 9893 with the choice as a hypothesis. (Contributed by Mario Carneiro, 27-Aug-2015.)
(( 𝐴 ∈ dom card ∧ ¬ ∅ ∈ 𝐴) → ∃𝑓(𝑓:𝐴 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝑥))

Theoremondomen 9452 If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.)
((𝐴 ∈ On ∧ 𝐵𝐴) → 𝐵 ∈ dom card)

Theoremnumdom 9453 A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)

Theoremssnum 9454 A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)

Theoremonssnum 9455 All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.)
((𝐴𝑉𝐴 ⊆ On) → 𝐴 ∈ dom card)

Theoremindcardi 9456* Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)
(𝜑𝐴𝑉)    &   (𝜑𝑇 ∈ dom card)    &   ((𝜑𝑅𝑇 ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)    &   (𝑥 = 𝑦 → (𝜓𝜒))    &   (𝑥 = 𝐴 → (𝜓𝜃))    &   (𝑥 = 𝑦𝑅 = 𝑆)    &   (𝑥 = 𝐴𝑅 = 𝑇)       (𝜑𝜃)

Theoremacnrcl 9457 Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑋AC 𝐴𝐴 ∈ V)

Theoremacneq 9458 Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 = 𝐶AC 𝐴 = AC 𝐶)

Theoremisacn 9459* The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
((𝑋𝑉𝐴𝑊) → (𝑋AC 𝐴 ↔ ∀𝑓 ∈ ((𝒫 𝑋 ∖ {∅}) ↑m 𝐴)∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥)))

Theoremacni 9460* The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
((𝑋AC 𝐴𝐹:𝐴⟶(𝒫 𝑋 ∖ {∅})) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝐹𝑥))

Theoremacni2 9461* The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
((𝑋AC 𝐴 ∧ ∀𝑥𝐴 (𝐵𝑋𝐵 ≠ ∅)) → ∃𝑔(𝑔:𝐴𝑋 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ 𝐵))

Theoremacni3 9462* The property of being a choice set of length 𝐴. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑦 = (𝑔𝑥) → (𝜑𝜓))       ((𝑋AC 𝐴 ∧ ∀𝑥𝐴𝑦𝑋 𝜑) → ∃𝑔(𝑔:𝐴𝑋 ∧ ∀𝑥𝐴 𝜓))

Theoremacnlem 9463* Construct a mapping satisfying the consequent of isacn 9459. (Contributed by Mario Carneiro, 31-Aug-2015.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))

Theoremnumacn 9464 A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴𝑉 → (𝑋 ∈ dom card → 𝑋AC 𝐴))

Theoremfinacn 9465 Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 ∈ Fin → AC 𝐴 = V)

Theoremacndom 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.)
(𝐴𝐵 → (𝑋AC 𝐵𝑋AC 𝐴))

Theoremacnnum 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.)
(𝑋AC 𝒫 𝑋𝑋 ∈ dom card)

Theoremacnen 9468 The class of choice sets of length 𝐴 is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴𝐵AC 𝐴 = AC 𝐵)

Theoremacndom2 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.)
(𝑋𝑌 → (𝑌AC 𝐴𝑋AC 𝐴))

Theoremacnen2 9470 The class of sets with choice sequences of length 𝐴 is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑋𝑌 → (𝑋AC 𝐴𝑌AC 𝐴))

Theoremfodomacn 9471 A version of fodom 9938 that doesn't require the Axiom of Choice ax-ac 9874. 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.)
(𝐴AC 𝐵 → (𝐹:𝐴onto𝐵𝐵𝐴))

Theoremfodomnum 9472 A version of fodom 9938 that doesn't require the Axiom of Choice ax-ac 9874. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐴 ∈ dom card → (𝐹:𝐴onto𝐵𝐵𝐴))

Theoremfonum 9473 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐹:𝐴onto𝐵) → 𝐵 ∈ dom card)

Theoremnumwdom 9474 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.)
((𝐴 ∈ dom card ∧ 𝐵* 𝐴) → 𝐵 ∈ dom card)

Theoremfodomfi2 9475 Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐴𝑉𝐵 ∈ Fin ∧ 𝐹:𝐴onto𝐵) → 𝐵𝐴)

Theoremwdomfil 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* 𝑌𝑋𝑌))

Theoreminfpwfien 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)

Theoreminffien 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)

Theoremwdomnumr 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* 𝐵𝐴𝐵))

Theoremalephfnon 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

Theoremaleph0 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 0. 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

Theoremalephlim 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.)
((𝐴𝑉 ∧ Lim 𝐴) → (ℵ‘𝐴) = 𝑥𝐴 (ℵ‘𝑥))

Theoremalephsuc 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 9009, 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)))

Theoremalephon 9484 An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
(ℵ‘𝐴) ∈ On

Theoremalephcard 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)

Theoremalephnbtwn 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) ∈ 𝐵𝐵 ∈ (ℵ‘suc 𝐴)))

Theoremalephnbtwn2 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) ≺ 𝐵𝐵 ≺ (ℵ‘suc 𝐴))

Theoremalephordilem1 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))

Theoremalephordi 9489 Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)
(𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

Theoremalephord 9490 Ordering property of the aleph function. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ≺ (ℵ‘𝐵)))

Theoremalephord2 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) → (𝐴𝐵 ↔ (ℵ‘𝐴) ∈ (ℵ‘𝐵)))

Theoremalephord2i 9492 Ordering property of the aleph function. Theorem 66 of [Suppes] p. 229. (Contributed by NM, 25-Oct-2003.)
(𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ∈ (ℵ‘𝐵)))

Theoremalephord3 9493 Ordering property of the aleph function. (Contributed by NM, 11-Nov-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ⊆ (ℵ‘𝐵)))

Theoremalephsucdom 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)))

Theoremalephsuc2 9495* An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 8996 function by transfinite recursion, starting from ω. Using this theorem we could define the aleph function with {𝑧 ∈ On ∣ 𝑧𝑥} in place of {𝑧 ∈ On ∣ 𝑥𝑧} in df-aleph 9357. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
(𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})

Theoremalephdom 9496 Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ≼ (ℵ‘𝐵)))

Theoremalephgeom 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))

Theoremalephislim 9498 Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.)
(𝐴 ∈ On ↔ Lim (ℵ‘𝐴))

Theoremaleph11 9499 The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) = (ℵ‘𝐵) ↔ 𝐴 = 𝐵))

Theoremalephf1 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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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45330
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