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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fz0n 35001 | The sequence (0...(π β 1)) is empty iff π is zero. (Contributed by Scott Fenton, 16-May-2014.) |
β’ (π β β0 β ((0...(π β 1)) = β β π = 0)) | ||
Theorem | shftvalg 35002 | Value of a sequence shifted by π΄. (Contributed by Scott Fenton, 16-Dec-2017.) |
β’ ((πΉ β π β§ π΄ β β β§ π΅ β β) β ((πΉ shift π΄)βπ΅) = (πΉβ(π΅ β π΄))) | ||
Theorem | divcnvlin 35003* | Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β π΄ β β) & β’ (π β π΅ β β€) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) = ((π + π΄) / (π + π΅))) β β’ (π β πΉ β 1) | ||
Theorem | climlec3 35004* | Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β π΅ β β) & β’ (π β πΉ β π΄) & β’ ((π β§ π β π) β (πΉβπ) β β) & β’ ((π β§ π β π) β (πΉβπ) β€ π΅) β β’ (π β π΄ β€ π΅) | ||
Theorem | logi 35005 | Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.) |
β’ (logβi) = (i Β· (Ο / 2)) | ||
Theorem | iexpire 35006 | i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.) |
β’ (iβπi) β β | ||
Theorem | bcneg1 35007 | The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.) |
β’ (π β β0 β (πC-1) = 0) | ||
Theorem | bcm1nt 35008 | The proportion of one bionmial coefficient to another with π decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.) |
β’ ((π β β β§ πΎ β (0...(π β 1))) β (πCπΎ) = (((π β 1)CπΎ) Β· (π / (π β πΎ)))) | ||
Theorem | bcprod 35009* | A product identity for binomial coefficents. (Contributed by Scott Fenton, 23-Jun-2020.) |
β’ (π β β β βπ β (1...(π β 1))((π β 1)Cπ) = βπ β (1...(π β 1))(πβ((2 Β· π) β π))) | ||
Theorem | bccolsum 35010* | A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.) |
β’ ((π β β0 β§ πΆ β β0) β Ξ£π β (0...π)(πCπΆ) = ((π + 1)C(πΆ + 1))) | ||
Theorem | iprodefisumlem 35011 | Lemma for iprodefisum 35012. (Contributed by Scott Fenton, 11-Feb-2018.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ:πβΆβ) β β’ (π β seqπ( Β· , (exp β πΉ)) = (exp β seqπ( + , πΉ))) | ||
Theorem | iprodefisum 35012* | Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ ((π β§ π β π) β (πΉβπ) = π΅) & β’ ((π β§ π β π) β π΅ β β) & β’ (π β seqπ( + , πΉ) β dom β ) β β’ (π β βπ β π (expβπ΅) = (expβΞ£π β π π΅)) | ||
Theorem | iprodgam 35013* | An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.) |
β’ (π β π΄ β (β β (β€ β β))) β β’ (π β (Ξβπ΄) = (βπ β β (((1 + (1 / π))βππ΄) / (1 + (π΄ / π))) / π΄)) | ||
Theorem | faclimlem1 35014* | Lemma for faclim 35017. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.) |
β’ (π β β0 β seq1( Β· , (π β β β¦ (((1 + (π / π)) Β· (1 + (1 / π))) / (1 + ((π + 1) / π))))) = (π₯ β β β¦ ((π + 1) Β· ((π₯ + 1) / (π₯ + (π + 1)))))) | ||
Theorem | faclimlem2 35015* | Lemma for faclim 35017. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.) |
β’ (π β β0 β seq1( Β· , (π β β β¦ (((1 + (π / π)) Β· (1 + (1 / π))) / (1 + ((π + 1) / π))))) β (π + 1)) | ||
Theorem | faclimlem3 35016 | Lemma for faclim 35017. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.) |
β’ ((π β β0 β§ π΅ β β) β (((1 + (1 / π΅))β(π + 1)) / (1 + ((π + 1) / π΅))) = ((((1 + (1 / π΅))βπ) / (1 + (π / π΅))) Β· (((1 + (π / π΅)) Β· (1 + (1 / π΅))) / (1 + ((π + 1) / π΅))))) | ||
Theorem | faclim 35017* | An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) |
β’ πΉ = (π β β β¦ (((1 + (1 / π))βπ΄) / (1 + (π΄ / π)))) β β’ (π΄ β β0 β seq1( Β· , πΉ) β (!βπ΄)) | ||
Theorem | iprodfac 35018* | An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.) |
β’ (π΄ β β0 β (!βπ΄) = βπ β β (((1 + (1 / π))βπ΄) / (1 + (π΄ / π)))) | ||
Theorem | faclim2 35019* | Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.) |
β’ πΉ = (π β β β¦ (((!βπ) Β· ((π + 1)βπ)) / (!β(π + π)))) β β’ (π β β0 β πΉ β 1) | ||
Theorem | gcd32 35020 | Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
β’ ((π΄ β β€ β§ π΅ β β€ β§ πΆ β β€) β ((π΄ gcd π΅) gcd πΆ) = ((π΄ gcd πΆ) gcd π΅)) | ||
Theorem | gcdabsorb 35021 | Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
β’ ((π΄ β β€ β§ π΅ β β€) β ((π΄ gcd π΅) gcd π΅) = (π΄ gcd π΅)) | ||
Theorem | dftr6 35022 | A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.) |
β’ π΄ β V β β’ (Tr π΄ β π΄ β (V β ran (( E β E ) β E ))) | ||
Theorem | coep 35023* | Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
β’ π΄ β V & β’ π΅ β V β β’ (π΄( E β π )π΅ β βπ₯ β π΅ π΄π π₯) | ||
Theorem | coepr 35024* | Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
β’ π΄ β V & β’ π΅ β V β β’ (π΄(π β β‘ E )π΅ β βπ₯ β π΄ π₯π π΅) | ||
Theorem | dffr5 35025 | A quantifier-free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.) |
β’ (π Fr π΄ β (π« π΄ β {β }) β ran ( E β ( E β β‘π ))) | ||
Theorem | dfso2 35026 | Quantifier-free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.) |
β’ (π Or π΄ β (π Po π΄ β§ (π΄ Γ π΄) β (π βͺ ( I βͺ β‘π )))) | ||
Theorem | br8 35027* | Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
β’ (π = π΄ β (π β π)) & β’ (π = π΅ β (π β π)) & β’ (π = πΆ β (π β π)) & β’ (π = π· β (π β π)) & β’ (π = πΈ β (π β π)) & β’ (π = πΉ β (π β π)) & β’ (π = πΊ β (π β π)) & β’ (β = π» β (π β π)) & β’ (π₯ = π β π = π) & β’ π = {β¨π, πβ© β£ βπ₯ β π βπ β π βπ β π βπ β π βπ β π βπ β π βπ β π βπ β π ββ β π (π = β¨β¨π, πβ©, β¨π, πβ©β© β§ π = β¨β¨π, πβ©, β¨π, ββ©β© β§ π)} β β’ (((π β π β§ π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π β§ πΈ β π) β§ (πΉ β π β§ πΊ β π β§ π» β π)) β (β¨β¨π΄, π΅β©, β¨πΆ, π·β©β©π β¨β¨πΈ, πΉβ©, β¨πΊ, π»β©β© β π)) | ||
Theorem | br6 35028* | Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
β’ (π = π΄ β (π β π)) & β’ (π = π΅ β (π β π)) & β’ (π = πΆ β (π β π)) & β’ (π = π· β (π β π)) & β’ (π = πΈ β (π β π)) & β’ (π = πΉ β (π β π)) & β’ (π₯ = π β π = π) & β’ π = {β¨π, πβ© β£ βπ₯ β π βπ β π βπ β π βπ β π βπ β π βπ β π βπ β π (π = β¨π, β¨π, πβ©β© β§ π = β¨π, β¨π, πβ©β© β§ π)} β β’ ((π β π β§ (π΄ β π β§ π΅ β π β§ πΆ β π) β§ (π· β π β§ πΈ β π β§ πΉ β π)) β (β¨π΄, β¨π΅, πΆβ©β©π β¨π·, β¨πΈ, πΉβ©β© β π)) | ||
Theorem | br4 35029* | Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.) |
β’ (π = π΄ β (π β π)) & β’ (π = π΅ β (π β π)) & β’ (π = πΆ β (π β π)) & β’ (π = π· β (π β π)) & β’ (π₯ = π β π = π) & β’ π = {β¨π, πβ© β£ βπ₯ β π βπ β π βπ β π βπ β π βπ β π (π = β¨π, πβ© β§ π = β¨π, πβ© β§ π)} β β’ ((π β π β§ (π΄ β π β§ π΅ β π) β§ (πΆ β π β§ π· β π)) β (β¨π΄, π΅β©π β¨πΆ, π·β© β π)) | ||
Theorem | cnvco1 35030 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
β’ β‘(β‘π΄ β π΅) = (β‘π΅ β π΄) | ||
Theorem | cnvco2 35031 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
β’ β‘(π΄ β β‘π΅) = (π΅ β β‘π΄) | ||
Theorem | eldm3 35032 | Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.) |
β’ (π΄ β dom π΅ β (π΅ βΎ {π΄}) β β ) | ||
Theorem | elrn3 35033 | Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.) |
β’ (π΄ β ran π΅ β (π΅ β© (V Γ {π΄})) β β ) | ||
Theorem | pocnv 35034 | The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
β’ (π Po π΄ β β‘π Po π΄) | ||
Theorem | socnv 35035 | The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
β’ (π Or π΄ β β‘π Or π΄) | ||
Theorem | sotrd 35036 | Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.) |
β’ (π β π Or π΄) & β’ (π β π β π΄) & β’ (π β π β π΄) & β’ (π β π β π΄) & β’ (π β ππ π) & β’ (π β ππ π) β β’ (π β ππ π) | ||
Theorem | elintfv 35037* | Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.) |
β’ π β V β β’ ((πΉ Fn π΄ β§ π΅ β π΄) β (π β β© (πΉ β π΅) β βπ¦ β π΅ π β (πΉβπ¦))) | ||
Theorem | funpsstri 35038 | A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.) |
β’ ((Fun π» β§ (πΉ β π» β§ πΊ β π») β§ (dom πΉ β dom πΊ β¨ dom πΊ β dom πΉ)) β (πΉ β πΊ β¨ πΉ = πΊ β¨ πΊ β πΉ)) | ||
Theorem | fundmpss 35039 | If a class πΉ is a proper subset of a function πΊ, then dom πΉ β dom πΊ. (Contributed by Scott Fenton, 20-Apr-2011.) |
β’ (Fun πΊ β (πΉ β πΊ β dom πΉ β dom πΊ)) | ||
Theorem | funsseq 35040 | 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.) |
β’ ((Fun πΉ β§ Fun πΊ β§ dom πΉ = dom πΊ) β (πΉ = πΊ β πΉ β πΊ)) | ||
Theorem | fununiq 35041 | The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
β’ π΄ β V & β’ π΅ β V & β’ πΆ β V β β’ (Fun πΉ β ((π΄πΉπ΅ β§ π΄πΉπΆ) β π΅ = πΆ)) | ||
Theorem | funbreq 35042 | An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.) |
β’ π΄ β V & β’ π΅ β V & β’ πΆ β V β β’ ((Fun πΉ β§ π΄πΉπ΅) β (π΄πΉπΆ β π΅ = πΆ)) | ||
Theorem | br1steq 35043 | Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
β’ π΄ β V & β’ π΅ β V β β’ (β¨π΄, π΅β©1st πΆ β πΆ = π΄) | ||
Theorem | br2ndeq 35044 | Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.) |
β’ π΄ β V & β’ π΅ β V β β’ (β¨π΄, π΅β©2nd πΆ β πΆ = π΅) | ||
Theorem | dfdm5 35045 | Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
β’ dom π΄ = ((1st βΎ (V Γ V)) β π΄) | ||
Theorem | dfrn5 35046 | Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
β’ ran π΄ = ((2nd βΎ (V Γ V)) β π΄) | ||
Theorem | opelco3 35047 | Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.) |
β’ (β¨π΄, π΅β© β (πΆ β π·) β π΅ β (πΆ β (π· β {π΄}))) | ||
Theorem | elima4 35048 | Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.) |
β’ (π΄ β (π β π΅) β (π β© (π΅ Γ {π΄})) β β ) | ||
Theorem | fv1stcnv 35049 | The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
β’ ((π β π΄ β§ π β π) β (β‘(1st βΎ (π΄ Γ {π}))βπ) = β¨π, πβ©) | ||
Theorem | fv2ndcnv 35050 | The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.) |
β’ ((π β π β§ π β π΄) β (β‘(2nd βΎ ({π} Γ π΄))βπ) = β¨π, πβ©) | ||
Theorem | setinds 35051* | Principle of set induction (or E-induction). If a property passes from all elements of π₯ to π₯ itself, then it holds for all π₯. (Contributed by Scott Fenton, 10-Mar-2011.) |
β’ (βπ¦ β π₯ [π¦ / π₯]π β π) β β’ π | ||
Theorem | setinds2f 35052* | E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
β’ β²π₯π & β’ (π₯ = π¦ β (π β π)) & β’ (βπ¦ β π₯ π β π) β β’ π | ||
Theorem | setinds2 35053* | E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) |
β’ (π₯ = π¦ β (π β π)) & β’ (βπ¦ β π₯ π β π) β β’ π | ||
Theorem | elpotr 35054* | A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.) |
β’ (βπ§ β π΄ Tr π§ β E Po π΄) | ||
Theorem | dford5reg 35055 | Given ax-reg 9590, an ordinal is a transitive class totally ordered by the membership relation. (Contributed by Scott Fenton, 28-Jan-2011.) |
β’ (Ord π΄ β (Tr π΄ β§ E Or π΄)) | ||
Theorem | dfon2lem1 35056 | Lemma for dfon2 35065. (Contributed by Scott Fenton, 28-Feb-2011.) |
β’ Tr βͺ {π₯ β£ (π β§ Tr π₯ β§ π)} | ||
Theorem | dfon2lem2 35057* | Lemma for dfon2 35065. (Contributed by Scott Fenton, 28-Feb-2011.) |
β’ βͺ {π₯ β£ (π₯ β π΄ β§ π β§ π)} β π΄ | ||
Theorem | dfon2lem3 35058* | Lemma for dfon2 35065. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.) |
β’ (π΄ β π β (βπ₯((π₯ β π΄ β§ Tr π₯) β π₯ β π΄) β (Tr π΄ β§ βπ§ β π΄ Β¬ π§ β π§))) | ||
Theorem | dfon2lem4 35059* | Lemma for dfon2 35065. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.) |
β’ π΄ β V & β’ π΅ β V β β’ ((βπ₯((π₯ β π΄ β§ Tr π₯) β π₯ β π΄) β§ βπ¦((π¦ β π΅ β§ Tr π¦) β π¦ β π΅)) β (π΄ β π΅ β¨ π΅ β π΄)) | ||
Theorem | dfon2lem5 35060* | Lemma for dfon2 35065. Two sets satisfying the new definition also satisfy trichotomy with respect to β. (Contributed by Scott Fenton, 25-Feb-2011.) |
β’ π΄ β V & β’ π΅ β V β β’ ((βπ₯((π₯ β π΄ β§ Tr π₯) β π₯ β π΄) β§ βπ¦((π¦ β π΅ β§ Tr π¦) β π¦ β π΅)) β (π΄ β π΅ β¨ π΄ = π΅ β¨ π΅ β π΄)) | ||
Theorem | dfon2lem6 35061* | Lemma for dfon2 35065. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.) |
β’ ((Tr π β§ βπ₯ β π βπ§((π§ β π₯ β§ Tr π§) β π§ β π₯)) β βπ¦((π¦ β π β§ Tr π¦) β π¦ β π)) | ||
Theorem | dfon2lem7 35062* | Lemma for dfon2 35065. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.) |
β’ π΄ β V β β’ (βπ₯((π₯ β π΄ β§ Tr π₯) β π₯ β π΄) β (π΅ β π΄ β βπ¦((π¦ β π΅ β§ Tr π¦) β π¦ β π΅))) | ||
Theorem | dfon2lem8 35063* | Lemma for dfon2 35065. The intersection of a nonempty class π΄ of new ordinals is itself a new ordinal and is contained within π΄ (Contributed by Scott Fenton, 26-Feb-2011.) |
β’ ((π΄ β β β§ βπ₯ β π΄ βπ¦((π¦ β π₯ β§ Tr π¦) β π¦ β π₯)) β (βπ§((π§ β β© π΄ β§ Tr π§) β π§ β β© π΄) β§ β© π΄ β π΄)) | ||
Theorem | dfon2lem9 35064* | Lemma for dfon2 35065. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.) |
β’ (βπ₯ β π΄ βπ¦((π¦ β π₯ β§ Tr π¦) β π¦ β π₯) β E Fr π΄) | ||
Theorem | dfon2 35065* | 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.) |
β’ On = {π₯ β£ βπ¦((π¦ β π₯ β§ Tr π¦) β π¦ β π₯)} | ||
Theorem | rdgprc0 35066 | The value of the recursive definition generator at β when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
β’ (Β¬ πΌ β V β (rec(πΉ, πΌ)ββ ) = β ) | ||
Theorem | rdgprc 35067 | The value of the recursive definition generator when πΌ is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
β’ (Β¬ πΌ β V β rec(πΉ, πΌ) = rec(πΉ, β )) | ||
Theorem | dfrdg2 35068* | Alternate definition of the recursive function generator when πΌ is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
β’ (πΌ β π β rec(πΉ, πΌ) = βͺ {π β£ βπ₯ β On (π Fn π₯ β§ βπ¦ β π₯ (πβπ¦) = if(π¦ = β , πΌ, if(Lim π¦, βͺ (π β π¦), (πΉβ(πββͺ π¦)))))}) | ||
Theorem | dfrdg3 35069* | Generalization of dfrdg2 35068 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
β’ rec(πΉ, πΌ) = βͺ {π β£ βπ₯ β On (π Fn π₯ β§ βπ¦ β π₯ (πβπ¦) = if(π¦ = β , if(πΌ β V, πΌ, β ), if(Lim π¦, βͺ (π β π¦), (πΉβ(πββͺ π¦)))))} | ||
Theorem | axextdfeq 35070 | A version of ax-ext 2702 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
β’ βπ§((π§ β π₯ β π§ β π¦) β ((π§ β π¦ β π§ β π₯) β (π₯ β π€ β π¦ β π€))) | ||
Theorem | ax8dfeq 35071 | A version of ax-8 2107 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.) |
β’ βπ§((π§ β π₯ β π§ β π¦) β (π€ β π₯ β π€ β π¦)) | ||
Theorem | axextdist 35072 | ax-ext 2702 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ ((Β¬ βπ§ π§ = π₯ β§ Β¬ βπ§ π§ = π¦) β (βπ§(π§ β π₯ β π§ β π¦) β π₯ = π¦)) | ||
Theorem | axextbdist 35073 | axextb 2705 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ ((Β¬ βπ§ π§ = π₯ β§ Β¬ βπ§ π§ = π¦) β (π₯ = π¦ β βπ§(π§ β π₯ β π§ β π¦))) | ||
Theorem | 19.12b 35074* | Version of 19.12vv 2342 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.) |
β’ β²π¦π & β’ β²π₯π β β’ (βπ₯βπ¦(π β π) β βπ¦βπ₯(π β π)) | ||
Theorem | exnel 35075 | There is always a set not in π¦. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ βπ₯ Β¬ π₯ β π¦ | ||
Theorem | distel 35076 | Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5438 and elirrv 9594.) (Contributed by Scott Fenton, 15-Dec-2010.) |
β’ (Β¬ βπ¦ π¦ = π₯ β Β¬ βπ¦ Β¬ π₯ β π¦) | ||
Theorem | axextndbi 35077 | axextnd 10589 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.) |
β’ βπ§(π₯ = π¦ β (π§ β π₯ β π§ β π¦)) | ||
Theorem | hbntg 35078 | A more general form of hbnt 2289. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ (βπ₯(π β βπ₯π) β (Β¬ π β βπ₯ Β¬ π)) | ||
Theorem | hbimtg 35079 | A more general and closed form of hbim 2294. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ ((βπ₯(π β βπ₯π) β§ (π β βπ₯π)) β ((π β π) β βπ₯(π β π))) | ||
Theorem | hbaltg 35080 | A more general and closed form of hbal 2166. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ (βπ₯(π β βπ¦π) β (βπ₯π β βπ¦βπ₯π)) | ||
Theorem | hbng 35081 | A more general form of hbn 2290. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ (π β βπ₯π) β β’ (Β¬ π β βπ₯ Β¬ π) | ||
Theorem | hbimg 35082 | A more general form of hbim 2294. (Contributed by Scott Fenton, 13-Dec-2010.) |
β’ (π β βπ₯π) & β’ (π β βπ₯π) β β’ ((π β π) β βπ₯(π β π)) | ||
Syntax | cwsuc 35083 | Declare the syntax for well-founded successor. |
class wsuc(π , π΄, π) | ||
Syntax | cwlim 35084 | Declare the syntax for well-founded limit class. |
class WLim(π , π΄) | ||
Definition | df-wsuc 35085 | Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
β’ wsuc(π , π΄, π) = inf(Pred(β‘π , π΄, π), π΄, π ) | ||
Definition | df-wlim 35086* | Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
β’ WLim(π , π΄) = {π₯ β π΄ β£ (π₯ β inf(π΄, π΄, π ) β§ π₯ = sup(Pred(π , π΄, π₯), π΄, π ))} | ||
Theorem | wsuceq123 35087 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
β’ ((π = π β§ π΄ = π΅ β§ π = π) β wsuc(π , π΄, π) = wsuc(π, π΅, π)) | ||
Theorem | wsuceq1 35088 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
β’ (π = π β wsuc(π , π΄, π) = wsuc(π, π΄, π)) | ||
Theorem | wsuceq2 35089 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
β’ (π΄ = π΅ β wsuc(π , π΄, π) = wsuc(π , π΅, π)) | ||
Theorem | wsuceq3 35090 | Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
β’ (π = π β wsuc(π , π΄, π) = wsuc(π , π΄, π)) | ||
Theorem | nfwsuc 35091 | Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
β’ β²π₯π & β’ β²π₯π΄ & β’ β²π₯π β β’ β²π₯wsuc(π , π΄, π) | ||
Theorem | wlimeq12 35092 | Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
β’ ((π = π β§ π΄ = π΅) β WLim(π , π΄) = WLim(π, π΅)) | ||
Theorem | wlimeq1 35093 | Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) |
β’ (π = π β WLim(π , π΄) = WLim(π, π΄)) | ||
Theorem | wlimeq2 35094 | Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) |
β’ (π΄ = π΅ β WLim(π , π΄) = WLim(π , π΅)) | ||
Theorem | nfwlim 35095 | Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
β’ β²π₯π & β’ β²π₯π΄ β β’ β²π₯WLim(π , π΄) | ||
Theorem | elwlim 35096 | Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
β’ (π β WLim(π , π΄) β (π β π΄ β§ π β inf(π΄, π΄, π ) β§ π = sup(Pred(π , π΄, π), π΄, π ))) | ||
Theorem | wzel 35097 | The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
β’ ((π We π΄ β§ π Se π΄ β§ π΄ β β ) β inf(π΄, π΄, π ) β π΄) | ||
Theorem | wsuclem 35098* | Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.) |
β’ (π β π We π΄) & β’ (π β π Se π΄) & β’ (π β π β π) & β’ (π β βπ€ β π΄ ππ π€) β β’ (π β βπ₯ β π΄ (βπ¦ β Pred (β‘π , π΄, π) Β¬ π¦π π₯ β§ βπ¦ β π΄ (π₯π π¦ β βπ§ β Pred (β‘π , π΄, π)π§π π¦))) | ||
Theorem | wsucex 35099 | Existence theorem for well-founded successor. (Contributed by Scott Fenton, 16-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
β’ (π β π Or π΄) β β’ (π β wsuc(π , π΄, π) β V) | ||
Theorem | wsuccl 35100* | If π is a set with an π successor in π΄, then its well-founded successor is a member of π΄. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
β’ (π β π We π΄) & β’ (π β π Se π΄) & β’ (π β π β π) & β’ (π β βπ¦ β π΄ ππ π¦) β β’ (π β wsuc(π , π΄, π) β π΄) |
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