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Theorem List for Metamath Proof Explorer - 31401-31500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdftr6 31401 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
𝐴 ∈ V       (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Theoremcoep 31402* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)

Theoremcoepr 31403* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)

Theoremdffr5 31404 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 31405 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 31406 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 31407* 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 31408* 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 31409* 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 31410 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Theoremcnvco1 31411 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
(𝐴𝐵) = (𝐵𝐴)

Theoremcnvco2 31412 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
(𝐴𝐵) = (𝐵𝐴)

Theoremeldm3 31413 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
(𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Theoremelrn3 31414 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
(𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Theorempocnv 31415 The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 Po 𝐴𝑅 Po 𝐴)

Theoremsocnv 31416 The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 Or 𝐴𝑅 Or 𝐴)

Theoremsotrd 31417 Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝑋𝐴)    &   (𝜑𝑌𝐴)    &   (𝜑𝑍𝐴)    &   (𝜑𝑋𝑅𝑌)    &   (𝜑𝑌𝑅𝑍)       (𝜑𝑋𝑅𝑍)

Theoremsotr3 31418 Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍))

Theoremsoasym 31419 Asymmetry law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.)
((𝑅 Or 𝐴 ∧ (𝑋𝐴𝑌𝐴)) → (𝑋𝑅𝑌 → ¬ 𝑌𝑅𝑋))

20.8.10  Properties of functions and mappings

Theoremfunpsstri 31420 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 31421 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 31422 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 31423 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 31424 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))

Theoremfunbreq 31425 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))

Theoremfprb 31426* 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 31427 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 31428 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 31429 Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))

Theorembr2ndeqg 31430 Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))

Theoremdfdm5 31431 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 31432 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 31433 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 31434 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 31435 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 31436 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⟩)

20.8.11  Epsilon induction

Theoremsetinds 31437* 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 31438* 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 31439* 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 31440* A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.)
(∀𝑧𝐴 Tr 𝑧 → E Po 𝐴)

Theoremdford5reg 31441 Given ax-reg 8457, 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 31442 Lemma for dfon2 31451. (Contributed by Scott Fenton, 28-Feb-2011.)
Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}

Theoremdfon2lem2 31443* Lemma for dfon2 31451. (Contributed by Scott Fenton, 28-Feb-2011.)
{𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴

Theoremdfon2lem3 31444* Lemma for dfon2 31451. 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 31445* Lemma for dfon2 31451. 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 31446* Lemma for dfon2 31451. 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 31447* Lemma for dfon2 31451. 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 31448* Lemma for dfon2 31451. 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 31449* Lemma for dfon2 31451. 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 31450* Lemma for dfon2 31451. 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 31451* 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 31452 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
dom E = V

Theoremrdgprc0 31453 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 31454 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 31455* 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 31456* Generalization of dfrdg2 31455 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 31457 A version of ax-ext 2601 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))

Theoremax8dfeq 31458 A version of ax-8 1989 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))

Theoremaxextdist 31459 ax-ext 2601 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 31460 axext4 2605 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 31461* Version of 19.12vv 2179 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 31462 There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
𝑥 ¬ 𝑥𝑦

Theoremdistel 31463 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4817 and elirrv 8464.) (Contributed by Scott Fenton, 15-Dec-2010.)
(¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)

Theoremaxextndbi 31464 axextnd 9373 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
𝑧(𝑥 = 𝑦 ↔ (𝑧𝑥𝑧𝑦))

20.8.14  Hypothesis builders

Theoremhbntg 31465 A more general form of hbnt 2140. (Contributed by Scott Fenton, 13-Dec-2010.)
(∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Theoremhbimtg 31466 A more general and closed form of hbim 2123. (Contributed by Scott Fenton, 13-Dec-2010.)
((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒𝜓) → ∀𝑥(𝜑𝜃)))

Theoremhbaltg 31467 A more general and closed form of hbal 2033. (Contributed by Scott Fenton, 13-Dec-2010.)
(∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦𝑥𝜓))

Theoremhbng 31468 A more general form of hbn 2142. (Contributed by Scott Fenton, 13-Dec-2010.)
(𝜑 → ∀𝑥𝜓)       𝜓 → ∀𝑥 ¬ 𝜑)

Theoremhbimg 31469 A more general form of hbim 2123. (Contributed by Scott Fenton, 13-Dec-2010.)
(𝜑 → ∀𝑥𝜓)    &   (𝜒 → ∀𝑥𝜃)       ((𝜓𝜒) → ∀𝑥(𝜑𝜃))

20.8.15  (Trans)finite Recursion Theorems

Theoremtfisg 31470* A closed form of tfis 7016. (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 31471 Define the transitive predecessor class as a class.
class TrPred(𝑅, 𝐴, 𝑋)

Definitiondf-trpred 31472* 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 31484 and trpredmintr 31485). 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 31473* 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 31474 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑅 = 𝑆 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Theoremtrpredeq2 31475 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝐴 = 𝐵 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Theoremtrpredeq3 31476 Equality theorem for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝑋 = 𝑌 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Theoremtrpredeq1d 31477 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝑅 = 𝑆)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑆, 𝐴, 𝑋))

Theoremtrpredeq2d 31478 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐵, 𝑋))

Theoremtrpredeq3d 31479 Equality deduction for transitive predecessors. (Contributed by Scott Fenton, 2-Feb-2011.)
(𝜑𝑋 = 𝑌)       (𝜑 → TrPred(𝑅, 𝐴, 𝑋) = TrPred(𝑅, 𝐴, 𝑌))

Theoremeltrpred 31480* 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 31482 and trpredmintr 31485. (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))

Theoremtrpredlem1 31481* Technical lemma for transitive predecessors properties. All values of the transitive predecessors' underlying function are subsets of the base set. (Contributed by Scott Fenton, 28-Apr-2012.)
(Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)

Theoremtrpredpred 31482 Assuming it exists, the predecessor class is a subset of the transitive predecessors. (Contributed by Scott Fenton, 18-Feb-2011.)
(Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑋) ⊆ TrPred(𝑅, 𝐴, 𝑋))

Theoremtrpredss 31483 The transitive predecessors form a subset of the base class. (Contributed by Scott Fenton, 20-Feb-2011.)
(Pred(𝑅, 𝐴, 𝑋) ∈ 𝐵 → TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐴)

Theoremtrpredtr 31484 The transitive predecessors are transitive in 𝑅 and 𝐴 (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Theoremtrpredmintr 31485* The transitive predecessors form the smallest class transitive in 𝑅 and 𝐴. That is, if 𝐵 is another 𝑅, 𝐴 transitive class containing Pred(𝑅, 𝐴, 𝑋), then TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐵 (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑋𝐴𝑅 Se 𝐴) ∧ (∀𝑦𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)

Theoremtrpredelss 31486 Given a transitive predecessor 𝑌 of 𝑋, the transitive predecessors of 𝑌 are a subset of the transitive predecessors of 𝑋. (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → TrPred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Theoremdftrpred3g 31487* The transitive predecessors of 𝑋 are equal to the predecessors of 𝑋 together with their transitive predecessors. (Contributed by Scott Fenton, 26-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = (Pred(𝑅, 𝐴, 𝑋) ∪ 𝑦 ∈ Pred (𝑅, 𝐴, 𝑋)TrPred(𝑅, 𝐴, 𝑦)))

Theoremdftrpred4g 31488* Another recursive expression for the transitive predecessors. (Contributed by Scott Fenton, 27-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = 𝑦 ∈ Pred (𝑅, 𝐴, 𝑋)({𝑦} ∪ TrPred(𝑅, 𝐴, 𝑦)))

Theoremtrpredpo 31489 If 𝑅 partially orders 𝐴, then the transitive predecessors are the same as the immediate predecessors . (Contributed by Scott Fenton, 28-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑅 Po 𝐴𝑋𝐴𝑅 Se 𝐴) → TrPred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑋))

Theoremtrpred0 31490 The class of transitive predecessors is empty when 𝐴 is empty. (Contributed by Scott Fenton, 30-Apr-2012.)
TrPred(𝑅, ∅, 𝑋) = ∅

Theoremtrpredex 31491 The transitive predecessors of a relation form a set (NOTE: this is the first theorem in the transitive predecessor series that requires infinity). (Contributed by Scott Fenton, 18-Feb-2011.)
TrPred(𝑅, 𝐴, 𝑋) ∈ V

Theoremtrpredrec 31492* If 𝑌 is an 𝑅, 𝐴 transitive predecessor, then it is either an immediate predecessor or there is a transitive predecessor between 𝑌 and 𝑋. (Contributed by Scott Fenton, 9-May-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))

20.8.17  Founded Induction

Theoremfrmin 31493* Every (possibly proper) subclass of a class 𝐴 with a founded, set-like relation 𝑅 has a minimal element. Lemma 4.3 of Don Monk's notes for Advanced Set Theory, which can be found at http://euclid.colorado.edu/~monkd/settheory. This is a very strong generalization of tz6.26 5680 and tz7.5 5713. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Theoremfrind 31494* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 31493). This principle states that if 𝐵 is a subclass of a founded class 𝐴 with the property that every element of 𝐵 whose initial segment is included in 𝐴 is itself equal to 𝐴. Compare wfi 5682 and tfi 7015, which are special cases of this theorem that do not require the axiom of infinity to prove. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵))) → 𝐴 = 𝐵)

Theoremfrindi 31495* The principle of founded induction. Theorem 4.4 of Don Monk's notes (see frmin 31493). This principle states that if 𝐵 is a subclass of a founded class 𝐴 with the property that every element of 𝐵 whose initial segment is included in 𝐴 is itself equal to 𝐴. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴       ((𝐵𝐴 ∧ ∀𝑦𝐴 (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵𝑦𝐵)) → 𝐴 = 𝐵)

Theoremfrinsg 31496* Founded Induction Schema. If a property passes from all elements less than 𝑦 of a founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))       ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)

Theoremfrins 31497* Founded Induction Schema. If a property passes from all elements less than 𝑦 of a founded class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))       (𝑦𝐴𝜑)

Theoremfrins2fg 31498* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 7-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))       ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)

Theoremfrins2f 31499* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 6-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑅 Fr 𝐴    &   𝑅 Se 𝐴    &   𝑦𝜓    &   (𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))       (𝑦𝐴𝜑)

Theoremfrins2g 31500* Founded Induction schema, using implicit substitution. (Contributed by Scott Fenton, 8-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
(𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓𝜑))    &   (𝑦 = 𝑧 → (𝜑𝜓))       ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)

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