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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)))

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