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Theorem List for Metamath Proof Explorer - 31601-31700   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.8.6  Infinite products

Theoremiprodefisumlem 31601 Lemma for iprodefisum 31602. (Contributed by Scott Fenton, 11-Feb-2018.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℂ)       (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Theoremiprodefisum 31602* Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → ∏𝑘𝑍 (exp‘𝐵) = (exp‘Σ𝑘𝑍 𝐵))

Theoremiprodgam 31603* An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))

20.8.7  Factorial limits

Theoremfaclimlem1 31604* Lemma for faclim 31607. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1))))))

Theoremfaclimlem2 31605* Lemma for faclim 31607. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1))

Theoremfaclimlem3 31606 Lemma for faclim 31607. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)
((𝑀 ∈ ℕ0𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵)))))

Theoremfaclim 31607* An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))       (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))

Theoremiprodfac 31608* An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))))

Theoremfaclim2 31609* Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀))))       (𝑀 ∈ ℕ0𝐹 ⇝ 1)

20.8.8  Greatest common divisor and divisibility

Theorempdivsq 31610 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ) → (𝑃𝑀𝑃 ∥ (𝑀↑2)))

Theoremdvdspw 31611 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐾𝑀𝐾 ∥ (𝑀𝑁)))

Theoremgcd32 31612 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵))

Theoremgcdabsorb 31613 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵))

20.8.9  Properties of relationships

Theorembrtp 31614 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
𝑋 ∈ V    &   𝑌 ∈ V       (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷) ∨ (𝑋 = 𝐸𝑌 = 𝐹)))

Theoremdftr6 31615 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
𝐴 ∈ V       (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Theoremcoep 31616* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)

Theoremcoepr 31617* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)

Theoremdffr5 31618 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
(𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))

Theoremdfso2 31619 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
(𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))))

Theoremdfpo2 31620 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
(𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅))

Theorembr8 31621* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
(𝑎 = 𝐴 → (𝜑𝜓))    &   (𝑏 = 𝐵 → (𝜓𝜒))    &   (𝑐 = 𝐶 → (𝜒𝜃))    &   (𝑑 = 𝐷 → (𝜃𝜏))    &   (𝑒 = 𝐸 → (𝜏𝜂))    &   (𝑓 = 𝐹 → (𝜂𝜁))    &   (𝑔 = 𝐺 → (𝜁𝜎))    &   ( = 𝐻 → (𝜎𝜌))    &   (𝑥 = 𝑋𝑃 = 𝑄)    &   𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑒𝑃𝑓𝑃𝑔𝑃𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ⟩⟩ ∧ 𝜑)}       (((𝑋𝑆𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄𝐸𝑄) ∧ (𝐹𝑄𝐺𝑄𝐻𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))

Theorembr6 31622* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
(𝑎 = 𝐴 → (𝜑𝜓))    &   (𝑏 = 𝐵 → (𝜓𝜒))    &   (𝑐 = 𝐶 → (𝜒𝜃))    &   (𝑑 = 𝐷 → (𝜃𝜏))    &   (𝑒 = 𝐸 → (𝜏𝜂))    &   (𝑓 = 𝐹 → (𝜂𝜁))    &   (𝑥 = 𝑋𝑃 = 𝑄)    &   𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑒𝑃𝑓𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)}       ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄𝐶𝑄) ∧ (𝐷𝑄𝐸𝑄𝐹𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁))

Theorembr4 31623* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
(𝑎 = 𝐴 → (𝜑𝜓))    &   (𝑏 = 𝐵 → (𝜓𝜒))    &   (𝑐 = 𝐶 → (𝜒𝜃))    &   (𝑑 = 𝐷 → (𝜃𝜏))    &   (𝑥 = 𝑋𝑃 = 𝑄)    &   𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}       ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜏))

Theoremdfres3 31624 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Theoremcnvco1 31625 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
(𝐴𝐵) = (𝐵𝐴)

Theoremcnvco2 31626 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
(𝐴𝐵) = (𝐵𝐴)

Theoremeldm3 31627 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
(𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Theoremelrn3 31628 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
(𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Theorempocnv 31629 The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 Po 𝐴𝑅 Po 𝐴)

Theoremsocnv 31630 The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 Or 𝐴𝑅 Or 𝐴)

Theoremsotrd 31631 Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝑍𝐴)    &   (𝜑𝑋𝑅𝑌)    &   (𝜑𝑌𝑅𝑍)       (𝜑𝑋𝑅𝑍)

Theoremsotr3 31632 Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Theoremsoasym 31633 Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))

Theoremsotrine 31634 Trichotomy law for strict orderings. (Contributed by Scott Fenton, 8-Dec-2021.)
((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝐶 ↔ (𝐵𝑅𝐶𝐶𝑅𝐵)))

Theoremeqfunresadj 31635 Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.)
(((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑌 ∈ dom 𝐹𝑌 ∈ dom 𝐺 ∧ (𝐹𝑌) = (𝐺𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌})))

Theoremeqfunressuc 31636 Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.)
(((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐺 ∧ (𝐹𝑋) = (𝐺𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))

Theoremfuneldmb 31637 If is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.)
((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ≠ ∅))

Theoremelintfv 31638* Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.)
𝑋 ∈ V       ((𝐹 Fn 𝐴𝐵𝐴) → (𝑋 (𝐹𝐵) ↔ ∀𝑦𝐵 𝑋 ∈ (𝐹𝑦)))

20.8.10  Properties of functions and mappings

Theoremfunpsstri 31639 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
((Fun 𝐻 ∧ (𝐹𝐻𝐺𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹𝐺𝐹 = 𝐺𝐺𝐹))

Theoremfundmpss 31640 If a class 𝐹 is a proper subset of a function 𝐺, then dom 𝐹 ⊊ dom 𝐺. (Contributed by Scott Fenton, 20-Apr-2011.)
(Fun 𝐺 → (𝐹𝐺 → dom 𝐹 ⊊ dom 𝐺))

Theoremfvresval 31641 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
(((𝐹𝐵)‘𝐴) = (𝐹𝐴) ∨ ((𝐹𝐵)‘𝐴) = ∅)

Theoremfunsseq 31642 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺𝐹𝐺))

Theoremfununiq 31643 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))

Theoremfunbreq 31644 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))

Theoremfprb 31645* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))

Theorembr1steq 31646 Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)

Theorembr2ndeq 31647 Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)

Theorembr1steqg 31648 Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))

Theorembr2ndeqg 31649 Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))

Theoremdfdm5 31650 Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)

Theoremdfrn5 31651 Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Theoremopelco3 31652 Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Theoremelima4 31653 Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.)
(𝐴 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅)

Theoremfv1stcnv 31654 The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝑋𝐴𝑌𝑉) → ((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Theoremfv2ndcnv 31655 The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝑋𝑉𝑌𝐴) → ((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)

Theoremimaindm 31656 The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.)
(𝑅𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))

20.8.11  Epsilon induction

Theoremsetinds 31657* Principle of E induction (set induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.)
(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑)       𝜑

Theoremsetinds2f 31658* E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (∀𝑦𝑥 𝜓𝜑)       𝜑

Theoremsetinds2 31659* E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (∀𝑦𝑥 𝜓𝜑)       𝜑

20.8.12  Ordinal numbers

Theoremelpotr 31660* A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.)
(∀𝑧𝐴 Tr 𝑧 → E Po 𝐴)

Theoremdford5reg 31661 Given ax-reg 8482, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))

Theoremdfon2lem1 31662 Lemma for dfon2 31671. (Contributed by Scott Fenton, 28-Feb-2011.)
Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}

Theoremdfon2lem2 31663* Lemma for dfon2 31671. (Contributed by Scott Fenton, 28-Feb-2011.)
{𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴

Theoremdfon2lem3 31664* Lemma for dfon2 31671. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
(𝐴𝑉 → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))

Theoremdfon2lem4 31665* Lemma for dfon2 31671. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐵𝐴))

Theoremdfon2lem5 31666* Lemma for dfon2 31671. Two sets satisfying the new definition also satisfy trichotomy with respect to . (Contributed by Scott Fenton, 25-Feb-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Theoremdfon2lem6 31667* Lemma for dfon2 31671. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)
((Tr 𝑆 ∧ ∀𝑥𝑆𝑧((𝑧𝑥 ∧ Tr 𝑧) → 𝑧𝑥)) → ∀𝑦((𝑦𝑆 ∧ Tr 𝑦) → 𝑦𝑆))

Theoremdfon2lem7 31668* Lemma for dfon2 31671. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)
𝐴 ∈ V       (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (𝐵𝐴 → ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)))

Theoremdfon2lem8 31669* Lemma for dfon2 31671. The intersection of a nonempty class 𝐴 of new ordinals is itself a new ordinal and is contained within 𝐴 (Contributed by Scott Fenton, 26-Feb-2011.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)) → (∀𝑧((𝑧 𝐴 ∧ Tr 𝑧) → 𝑧 𝐴) ∧ 𝐴𝐴))

Theoremdfon2lem9 31670* Lemma for dfon2 31671. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.)
(∀𝑥𝐴𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → E Fr 𝐴)

Theoremdfon2 31671* On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers," American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)
On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}

Theoremdomep 31672 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
dom E = V

Theoremrdgprc0 31673 The value of the recursive definition generator at when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Theoremrdgprc 31674 The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Theoremdfrdg2 31675* Alternate definition of the recursive function generator when 𝐼 is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐼𝑉 → rec(𝐹, 𝐼) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))})

Theoremdfrdg3 31676* Generalization of dfrdg2 31675 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
rec(𝐹, 𝐼) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}

20.8.13  Defined equality axioms

Theoremaxextdfeq 31677 A version of ax-ext 2600 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))

Theoremax8dfeq 31678 A version of ax-8 1990 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))

Theoremaxextdist 31679 ax-ext 2600 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))

Theoremaxext4dist 31680 axext4 2604 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))

Theorem19.12b 31681* Version of 19.12vv 2178 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))

Theoremexnel 31682 There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
𝑥 ¬ 𝑥𝑦

Theoremdistel 31683 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4838 and elirrv 8489.) (Contributed by Scott Fenton, 15-Dec-2010.)
(¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)

Theoremaxextndbi 31684 axextnd 9398 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
𝑧(𝑥 = 𝑦 ↔ (𝑧𝑥𝑧𝑦))

20.8.14  Hypothesis builders

Theoremhbntg 31685 A more general form of hbnt 2142. (Contributed by Scott Fenton, 13-Dec-2010.)
(∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Theoremhbimtg 31686 A more general and closed form of hbim 2125. (Contributed by Scott Fenton, 13-Dec-2010.)
((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒𝜓) → ∀𝑥(𝜑𝜃)))

Theoremhbaltg 31687 A more general and closed form of hbal 2034. (Contributed by Scott Fenton, 13-Dec-2010.)
(∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦𝑥𝜓))

Theoremhbng 31688 A more general form of hbn 2144. (Contributed by Scott Fenton, 13-Dec-2010.)
(𝜑 → ∀𝑥𝜓)       𝜓 → ∀𝑥 ¬ 𝜑)

Theoremhbimg 31689 A more general form of hbim 2125. (Contributed by Scott Fenton, 13-Dec-2010.)
(𝜑 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜃)       ((𝜓𝜒) → ∀𝑥(𝜑𝜃))

20.8.15  (Trans)finite Recursion Theorems

Theoremtfisg 31690* A closed form of tfis 7039. (Contributed by Scott Fenton, 8-Jun-2011.)
(∀𝑥 ∈ On (∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑) → ∀𝑥 ∈ On 𝜑)

20.8.16  Transitive closure under a relationship

Syntaxctrpred 31691 Define the transitive predecessor class as a class.
class TrPred(𝑅, 𝐴, 𝑋)

Definitiondf-trpred 31692* Define the transitive predecessors of a class 𝑋 under a relationship 𝑅 and a class 𝐴. This class can be thought of as the "smallest" class containing all elements of 𝐴 that are linked to 𝑋 by a chain of 𝑅 relationships (see trpredtr 31704 and trpredmintr 31705). Definition based off of Lemma 4.2 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory (check The Internet Archive for it now as Prof. Monk appears to have rewritten his website). (Contributed by Scott Fenton, 2-Feb-2011.)
TrPred(𝑅, 𝐴, 𝑋) = ran (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)

Theoremdftrpred2 31693* A definition of the transitive predecessors of a class in terms of indexed union. (Contributed by Scott Fenton, 28-Apr-2012.)
TrPred(𝑅, 𝐴, 𝑋) = 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)

Theoremtrpredeq1 31694 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 = 𝑆 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Theoremtrpredeq2 31695 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Theoremtrpredeq3 31696 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Theoremtrpredeq1d 31697 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝑅 = 𝑆)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Theoremtrpredeq2d 31698 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Theoremtrpredeq3d 31699 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝑋 = 𝑌)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Theoremeltrpred 31700* A class is a transitive predecessor iff it is in some value of the underlying function. This theorem is not really meant to be used directly: instead refer to trpredpred 31702 and trpredmintr 31705. (Contributed by Scott Fenton, 28-Apr-2012.)
(𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) ↔ ∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖))

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