HomeHome Metamath Proof Explorer
Theorem List (p. 87 of 425)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreminfxpenc2lem1 8601* Lemma for infxpenc2 8604. (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 8602* Lemma for infxpenc2 8604. (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 8603* Lemma for infxpenc2 8604. (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 8604* Existence form of infxpenc 8600. A "uniform" or "canonical" version of infxpen 8596, 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 8605* 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 8606* Lemma for fseqen 8609. (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 8607* Lemma for fseqen 8609. (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 8608* One half of fseqen 8609. (Contributed by Mario Carneiro, 18-Nov-2014.)
(𝐴𝑉 → (ω × 𝐴) ≼ 𝑛 ∈ ω (𝐴𝑚 𝑛))
 
Theoremfseqen 8609* 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 8610 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 8611* Lemma for dfac8a 8612. 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 8612* 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 8613* 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 8614* Lemma for dfac8c 8615. (Contributed by Mario Carneiro, 10-Jan-2013.)
𝐹 = (𝑠 ∈ (𝐴 ∖ {∅}) ↦ (𝑎𝑠𝑏𝑠 ¬ 𝑏𝑟𝑎))       (𝐴𝐵 → (∃𝑟 𝑟 We 𝐴 → ∃𝑓𝑧𝐴 (𝑧 ≠ ∅ → (𝑓𝑧) ∈ 𝑧)))
 
Theoremdfac8c 8615* 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 8616* A proof of the Well ordering theorem weth 9076, 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 8617* 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 8618* A version of ac5b 9059 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 8619 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 8620 A set dominated by a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
 
Theoremssnum 8621 A subset of a numerable set is numerable. (Contributed by Mario Carneiro, 28-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐵𝐴) → 𝐵 ∈ dom card)
 
Theoremonssnum 8622 All subsets of the ordinals are numerable. (Contributed by Mario Carneiro, 12-Feb-2013.)
((𝐴𝑉𝐴 ⊆ On) → 𝐴 ∈ dom card)
 
Theoremindcardi 8623* 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 8624 Reverse closure for the choice set predicate. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝑋AC 𝐴𝐴 ∈ V)
 
Theoremacneq 8625 Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 = 𝐶AC 𝐴 = AC 𝐶)
 
Theoremisacn 8626* 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 8627* 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 8628* 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 8629* 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 8630* Construct a mapping satisfying the consequent of isacn 8626. (Contributed by Mario Carneiro, 31-Aug-2015.)
((𝐴𝑉 ∧ ∀𝑥𝐴 𝐵 ∈ (𝑓𝑥)) → ∃𝑔𝑥𝐴 (𝑔𝑥) ∈ (𝑓𝑥))
 
Theoremnumacn 8631 A well-orderable set has choice sequences of every length. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴𝑉 → (𝑋 ∈ dom card → 𝑋AC 𝐴))
 
Theoremfinacn 8632 Every set has finite choice sequences. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴 ∈ Fin → AC 𝐴 = V)
 
Theoremacndom 8633 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 8634 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 8635 The class of choice sets of length 𝐴 is a cardinal invariant. (Contributed by Mario Carneiro, 31-Aug-2015.)
(𝐴𝐵AC 𝐴 = AC 𝐵)
 
Theoremacndom2 8636 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 8637 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 8638 A version of fodom 9103 that doesn't require the Axiom of Choice ax-ac 9040. 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 8639 A version of fodom 9103 that doesn't require the Axiom of Choice ax-ac 9040. (Contributed by Mario Carneiro, 28-Feb-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐴 ∈ dom card → (𝐹:𝐴onto𝐵𝐵𝐴))
 
Theoremfonum 8640 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 30-Apr-2015.)
((𝐴 ∈ dom card ∧ 𝐹:𝐴onto𝐵) → 𝐵 ∈ dom card)
 
Theoremnumwdom 8641 A surjection maps numerable sets to numerable sets. (Contributed by Mario Carneiro, 27-Aug-2015.)
((𝐴 ∈ dom card ∧ 𝐵* 𝐴) → 𝐵 ∈ dom card)
 
Theoremfodomfi2 8642 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 8643 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 8644 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 8645 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 8646 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 8647 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 8648 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 8649* 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 8650 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 8222, 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 8651 An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
(ℵ‘𝐴) ∈ On
 
Theoremalephcard 8652 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 8653 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 8654 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 8655 Lemma for alephordi 8656. (Contributed by NM, 23-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)
(𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
 
Theoremalephordi 8656 Strict ordering property of the aleph function. (Contributed by Mario Carneiro, 2-Feb-2013.)
(𝐵 ∈ On → (𝐴𝐵 → (ℵ‘𝐴) ≺ (ℵ‘𝐵)))
 
Theoremalephord 8657 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 8658 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 8659 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 8660 Ordering property of the aleph function. (Contributed by NM, 11-Nov-2003.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (ℵ‘𝐴) ⊆ (ℵ‘𝐵)))
 
Theoremalephsucdom 8661 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 8662* An alternate representation of a successor aleph. The aleph function is the function obtained from the hartogs 8208 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 8525. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
(𝐴 ∈ On → (ℵ‘suc 𝐴) = {𝑥 ∈ On ∣ 𝑥 ≼ (ℵ‘𝐴)})
 
Theoremalephdom 8663 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 8664 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 8665 Every aleph is a limit ordinal. (Contributed by NM, 11-Nov-2003.)
(𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
 
Theoremaleph11 8666 The aleph function is one-to-one. (Contributed by NM, 3-Aug-2004.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) = (ℵ‘𝐵) ↔ 𝐴 = 𝐵))
 
Theoremalephf1 8667 The aleph function is a one-to-one mapping from the ordinals to the infinite cardinals. See also alephf1ALT 8685. (Contributed by Mario Carneiro, 2-Feb-2013.)
ℵ:On–1-1→On
 
Theoremalephsdom 8668 If an ordinal is smaller than an initial ordinal, it is strictly dominated by it. (Contributed by Jeff Hankins, 24-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘𝐵)))
 
Theoremalephdom2 8669 A dominated initial ordinal is included. (Contributed by Jeff Hankins, 24-Oct-2009.)
((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((ℵ‘𝐴) ⊆ 𝐵 ↔ (ℵ‘𝐴) ≼ 𝐵))
 
Theoremalephle 8670 The argument of the aleph function is less than or equal to its value. Exercise 2 of [TakeutiZaring] p. 91. (Later, in alephfp2 8691, we will that equality can sometimes hold.) (Contributed by NM, 9-Nov-2003.) (Proof shortened by Mario Carneiro, 22-Feb-2013.)
(𝐴 ∈ On → 𝐴 ⊆ (ℵ‘𝐴))
 
Theoremcardaleph 8671* Given any transfinite cardinal number 𝐴, there is exactly one aleph that is equal to it. Here we compute that aleph explicitly. (Contributed by NM, 9-Nov-2003.) (Revised by Mario Carneiro, 2-Feb-2013.)
((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) → 𝐴 = (ℵ‘ {𝑥 ∈ On ∣ 𝐴 ⊆ (ℵ‘𝑥)}))
 
Theoremcardalephex 8672* Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.)
(ω ⊆ 𝐴 → ((card‘𝐴) = 𝐴 ↔ ∃𝑥 ∈ On 𝐴 = (ℵ‘𝑥)))
 
Theoreminfenaleph 8673* An infinite numerable set is equinumerous to an infinite initial ordinal. (Contributed by Jeff Hankins, 23-Oct-2009.) (Revised by Mario Carneiro, 29-Apr-2015.)
((𝐴 ∈ dom card ∧ ω ≼ 𝐴) → ∃𝑥 ∈ ran ℵ𝑥𝐴)
 
Theoremisinfcard 8674 Two ways to express the property of being a transfinite cardinal. (Contributed by NM, 9-Nov-2003.)
((ω ⊆ 𝐴 ∧ (card‘𝐴) = 𝐴) ↔ 𝐴 ∈ ran ℵ)
 
Theoremiscard3 8675 Two ways to express the property of being a cardinal number. (Contributed by NM, 9-Nov-2003.)
((card‘𝐴) = 𝐴𝐴 ∈ (ω ∪ ran ℵ))
 
Theoremcardnum 8676 Two ways to express the class of all cardinal numbers, which consists of the finite ordinals in ω plus the transfinite alephs. (Contributed by NM, 10-Sep-2004.)
{𝑥 ∣ (card‘𝑥) = 𝑥} = (ω ∪ ran ℵ)
 
Theoremalephinit 8677* An infinite initial ordinal is characterized by the property of being initial - that is, it is a subset of any dominating ordinal. (Contributed by Jeff Hankins, 29-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.)
((𝐴 ∈ On ∧ ω ⊆ 𝐴) → (𝐴 ∈ ran ℵ ↔ ∀𝑥 ∈ On (𝐴𝑥𝐴𝑥)))
 
Theoremcarduniima 8678 The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
(𝐴𝐵 → (𝐹:𝐴⟶(ω ∪ ran ℵ) → (𝐹𝐴) ∈ (ω ∪ ran ℵ)))
 
Theoremcardinfima 8679* If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of [TakeutiZaring] p. 104. (Contributed by NM, 4-Nov-2004.)
(𝐴𝐵 → ((𝐹:𝐴⟶(ω ∪ ran ℵ) ∧ ∃𝑥𝐴 (𝐹𝑥) ∈ ran ℵ) → (𝐹𝐴) ∈ ran ℵ))
 
Theoremalephiso 8680 Aleph is an order isomorphism of the class of ordinal numbers onto the class of infinite cardinals. Definition 10.27 of [TakeutiZaring] p. 90. (Contributed by NM, 3-Aug-2004.)
ℵ Isom E , E (On, {𝑥 ∣ (ω ⊆ 𝑥 ∧ (card‘𝑥) = 𝑥)})
 
Theoremalephprc 8681 The class of all transfinite cardinal numbers (the range of the aleph function) is a proper class. Proposition 10.26 of [TakeutiZaring] p. 90. (Contributed by NM, 11-Nov-2003.)
¬ ran ℵ ∈ V
 
Theoremalephsson 8682 The class of transfinite cardinals (the range of the aleph function) is a subclass of the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
ran ℵ ⊆ On
 
Theoremunialeph 8683 The union of the class of transfinite cardinals (the range of the aleph function) is the class of ordinal numbers. (Contributed by NM, 11-Nov-2003.)
ran ℵ = On
 
Theoremalephsmo 8684 The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
Smo ℵ
 
Theoremalephf1ALT 8685 Alternate proof of alephf1 8667. (Contributed by Mario Carneiro, 15-Mar-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
ℵ:On–1-1→On
 
Theoremalephfplem1 8686 Lemma for alephfp 8690. (Contributed by NM, 6-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (𝐻‘∅) ∈ ran ℵ
 
Theoremalephfplem2 8687* Lemma for alephfp 8690. (Contributed by NM, 6-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (𝑤 ∈ ω → (𝐻‘suc 𝑤) = (ℵ‘(𝐻𝑤)))
 
Theoremalephfplem3 8688* Lemma for alephfp 8690. (Contributed by NM, 6-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (𝑣 ∈ ω → (𝐻𝑣) ∈ ran ℵ)
 
Theoremalephfplem4 8689 Lemma for alephfp 8690. (Contributed by NM, 5-Nov-2004.)
𝐻 = (rec(ℵ, ω) ↾ ω)        (𝐻 “ ω) ∈ ran ℵ
 
Theoremalephfp 8690 The aleph function has a fixed point. Similar to Proposition 11.18 of [TakeutiZaring] p. 104, except that we construct an actual example of a fixed point rather than just showing its existence. See alephfp2 8691 for an abbreviated version just showing existence. (Contributed by NM, 6-Nov-2004.) (Proof shortened by Mario Carneiro, 15-May-2015.)
𝐻 = (rec(ℵ, ω) ↾ ω)       (ℵ‘ (𝐻 “ ω)) = (𝐻 “ ω)
 
Theoremalephfp2 8691 The aleph function has at least one fixed point. Proposition 11.18 of [TakeutiZaring] p. 104. See alephfp 8690 for an actual example of a fixed point. Compare the inequality alephle 8670 that holds in general. Note that if 𝑥 is a fixed point, then ℵ‘ℵ‘ℵ‘... ℵ‘𝑥 = 𝑥. (Contributed by NM, 6-Nov-2004.) (Revised by Mario Carneiro, 15-May-2015.)
𝑥 ∈ On (ℵ‘𝑥) = 𝑥
 
Theoremalephval3 8692* An alternate way to express the value of the aleph function: it is the least infinite cardinal different from all values at smaller arguments. Definition of aleph in [Enderton] p. 212 and definition of aleph in [BellMachover] p. 490 . (Contributed by NM, 16-Nov-2003.)
(𝐴 ∈ On → (ℵ‘𝐴) = {𝑥 ∣ ((card‘𝑥) = 𝑥 ∧ ω ⊆ 𝑥 ∧ ∀𝑦𝐴 ¬ 𝑥 = (ℵ‘𝑦))})
 
Theoremalephsucpw2 8693 The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 9253 or gchaleph2 9249.) The transposed form alephsucpw 9147 cannot be proven without the AC, and is in fact equivalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)
¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)
 
Theoremmappwen 8694 Power rule for cardinal arithmetic. Theorem 11.21 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)
(((𝐵 ∈ dom card ∧ ω ≼ 𝐵) ∧ (2𝑜𝐴𝐴 ≼ 𝒫 𝐵)) → (𝐴𝑚 𝐵) ≈ 𝒫 𝐵)
 
Theoremfinnisoeu 8695* A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.)
((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
 
Theoremiunfictbso 8696 Countability of a countable union of finite sets with a strict (not globally well) order fulfilling the choice role. (Contributed by Stefan O'Rear, 16-Nov-2014.)
((𝐴 ≼ ω ∧ 𝐴 ⊆ Fin ∧ 𝐵 Or 𝐴) → 𝐴 ≼ ω)
 
2.6.8  Axiom of Choice equivalents
 
Syntaxwac 8697 Wff for an abbreviation of the axiom of choice.
wff CHOICE
 
Definitiondf-ac 8698* The expression CHOICE will be used as a readable shorthand for any form of the axiom of choice; all concrete forms are long, cryptic, have dummy variables, or all three, making it useful to have a short name. Similar to the Axiom of Choice (first form) of [Enderton] p. 49.

There is a slight problem with taking the exact form of ax-ac 9040 as our definition, because the equivalence to more standard forms (dfac2 8712) requires the Axiom of Regularity, which we often try to avoid. Thus, we take the first of the "textbook forms" as the definition and derive the form of ax-ac 9040 itself as dfac0 8714. (Contributed by Mario Carneiro, 22-Feb-2015.)

(CHOICE ↔ ∀𝑥𝑓(𝑓𝑥𝑓 Fn dom 𝑥))
 
Theoremaceq1 8699* Equivalence of two versions of the Axiom of Choice ax-ac 9040. The proof uses neither AC nor the Axiom of Regularity. The right-hand side expresses our AC with the fewest number of different variables. (Contributed by NM, 5-Apr-2004.)
(∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑥𝑧(∃𝑥((𝑧𝑤𝑤𝑥) ∧ (𝑧𝑥𝑥𝑦)) ↔ 𝑧 = 𝑥)))
 
Theoremaceq0 8700* Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac 9040. (Contributed by NM, 5-Apr-2004.)
(∃𝑦𝑧𝑥𝑤𝑧 ∃!𝑣𝑧𝑢𝑦 (𝑧𝑢𝑣𝑢) ↔ ∃𝑦𝑧𝑤((𝑧𝑤𝑤𝑥) → ∃𝑣𝑢(∃𝑡((𝑢𝑤𝑤𝑡) ∧ (𝑢𝑡𝑡𝑦)) ↔ 𝑢 = 𝑣)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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 42401-42426
  Copyright terms: Public domain < Previous  Next >