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Theorem List for Intuitionistic Logic Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelpw2g 3901 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(B 𝑉 → (A 𝒫 BAB))
 
Theoremelpw2 3902 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
B V       (A 𝒫 BAB)
 
Theorempwnss 3903 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(A 𝑉 → ¬ 𝒫 AA)
 
Theorempwne 3904 No set equals its power set. The sethood antecedent is necessary; compare pwv 3570. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
(A 𝑉 → 𝒫 AA)
 
Theoremrepizf2lem 3905 Lemma for repizf2 3906. If we have a function-like proposition which provides at most one value of y for each x in a set w, we can change "at most one" to "exactly one" by restricting the values of x to those values for which the proposition provides a value of y. (Contributed by Jim Kingdon, 7-Sep-2018.)
(x w ∃*yφx {x wyφ}∃!yφ)
 
Theoremrepizf2 3906* Replacement. This version of replacement is stronger than repizf 3864 in the sense that φ does not need to map all values of x in w to a value of y. The resulting set contains those elements for which there is a value of y and in that sense, this theorem combines repizf 3864 with ax-sep 3866. Another variation would be x w∃*yφ → {yx(x w φ)} V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.)
zφ       (x w ∃*yφzx {x wyφ}y z φ)
 
2.2.5  Theorems requiring empty set existence
 
Theoremclass2seteq 3907* Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(A 𝑉 → {x AA V} = A)
 
Theorem0elpw 3908 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
𝒫 A
 
Theorem0nep0 3909 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}
 
Theorem0inp0 3910 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 30-Sep-2003.)
(A = ∅ → ¬ A = {∅})
 
Theoremunidif0 3911 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(A ∖ {∅}) = A
 
Theoremiin0imm 3912* An indexed intersection of the empty set, with an inhabited index set, is empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
(y y A x A ∅ = ∅)
 
Theoremiin0r 3913* If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.)
( x A ∅ = ∅ → A ≠ ∅)
 
Theoremintv 3914 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅
 
Theoremaxpweq 3915* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 3918 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
A V       (𝒫 A V ↔ xy(z(z yz A) → y x))
 
2.2.6  Collection principle
 
Theorembnd 3916* A very strong generalization of the Axiom of Replacement (compare zfrep6 3865). Its strength lies in the rather profound fact that φ(x, y) does not have to be a "function-like" wff, as it does in the standard Axiom of Replacement. This theorem is sometimes called the Boundedness Axiom. In the context of IZF, it is just a slight variation of ax-coll 3863. (Contributed by NM, 17-Oct-2004.)
(x z yφwx z y w φ)
 
Theorembnd2 3917* A variant of the Boundedness Axiom bnd 3916 that picks a subset z out of a possibly proper class B in which a property is true. (Contributed by NM, 4-Feb-2004.)
A V       (x A y B φz(zB x A y z φ))
 
2.3  IZF Set Theory - add the Axioms of Power Sets and Pairing
 
2.3.1  Introduce the Axiom of Power Sets
 
Axiomax-pow 3918* Axiom of Power Sets. An axiom of Intuitionistic Zermelo-Fraenkel set theory. It states that a set y exists that includes the power set of a given set x i.e. contains every subset of x. This is Axiom 8 of [Crosilla] p. "Axioms of CZF and IZF" except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3869).

The variant axpow2 3920 uses explicit subset notation. A version using class notation is pwex 3923. (Contributed by NM, 5-Aug-1993.)

yz(w(w zw x) → z y)
 
Theoremzfpow 3919* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
xy(x(x yx z) → y x)
 
Theoremaxpow2 3920* A variant of the Axiom of Power Sets ax-pow 3918 using subset notation. Problem in {BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
yz(zxz y)
 
Theoremaxpow3 3921* A variant of the Axiom of Power Sets ax-pow 3918. For any set x, there exists a set y whose members are exactly the subsets of x i.e. the power set of x. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
yz(zxz y)
 
Theoremel 3922* Every set is an element of some other set. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
y x y
 
Theorempwex 3923 Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
A V       𝒫 A V
 
Theorempwexg 3924 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)
(A 𝑉 → 𝒫 A V)
 
Theoremabssexg 3925* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(A 𝑉 → {x ∣ (xA φ)} V)
 
TheoremsnexgOLD 3926 A singleton whose element exists is a set. The A V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. This is a special case of snexg 3927 and new proofs should use snexg 3927 instead. (Contributed by Jim Kingdon, 26-Jan-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of snexg 3927 and then remove it.
(A V → {A} V)
 
Theoremsnexg 3927 A singleton whose element exists is a set. The A V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
(A 𝑉 → {A} V)
 
Theoremsnex 3928 A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)
A V       {A} V
 
Theoremsnexprc 3929 A singleton whose element is a proper class is a set. The ¬ A V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
A V → {A} V)
 
Theoremp0ex 3930 The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
{∅} V
 
Theorempp0ex 3931 {∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)
{∅, {∅}} V
 
Theoremord3ex 3932 The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
{∅, {∅}, {∅, {∅}}} V
 
Theoremdtruarb 3933* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). This theorem asserts the existence of two sets which do not equal each other; compare with dtruex 4237 in which we are given a set y and go from there to a set x which is not equal to it. (Contributed by Jim Kingdon, 2-Sep-2018.)
xy ¬ x = y
 
Theorempwuni 3934 A class is a subclass of the power class of its union. Exercise 6(b) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.)
A ⊆ 𝒫 A
 
2.3.2  Axiom of Pairing
 
Axiomax-pr 3935* The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 3869). (Contributed by NM, 14-Nov-2006.)
zw((w = x w = y) → w z)
 
Theoremzfpair2 3936 Derive the abbreviated version of the Axiom of Pairing from ax-pr 3935. (Contributed by NM, 14-Nov-2006.)
{x, y} V
 
TheoremprexgOLD 3937 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3471, prprc1 3469, and prprc2 3470. This is a special case of prexg 3938 and new proofs should use prexg 3938 instead. (Contributed by Jim Kingdon, 25-Jul-2019.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of prexg 3938 and then remove it.
((A V B V) → {A, B} V)
 
Theoremprexg 3938 The Axiom of Pairing using class variables. Theorem 7.13 of [Quine] p. 51, but restricted to classes which exist. For proper classes, see prprc 3471, prprc1 3469, and prprc2 3470. (Contributed by Jim Kingdon, 16-Sep-2018.)
((A 𝑉 B 𝑊) → {A, B} V)
 
Theoremsnelpwi 3939 A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011.)
(A B → {A} 𝒫 B)
 
Theoremsnelpw 3940 A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)
A V       (A B ↔ {A} 𝒫 B)
 
Theoremprelpwi 3941 A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)
((A 𝐶 B 𝐶) → {A, B} 𝒫 𝐶)
 
Theoremrext 3942* A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
(z(x zy z) → x = y)
 
Theoremsspwb 3943 Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
(AB ↔ 𝒫 A ⊆ 𝒫 B)
 
Theoremunipw 3944 A class equals the union of its power class. Exercise 6(a) of [Enderton] p. 38. (Contributed by NM, 14-Oct-1996.) (Proof shortened by Alan Sare, 28-Dec-2008.)
𝒫 A = A
 
Theorempwel 3945 Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
(A B → 𝒫 A 𝒫 𝒫 B)
 
Theorempwtr 3946 A class is transitive iff its power class is transitive. (Contributed by Alan Sare, 25-Aug-2011.) (Revised by Mario Carneiro, 15-Jun-2014.)
(Tr A ↔ Tr 𝒫 A)
 
Theoremssextss 3947* An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004.)
(ABx(xAxB))
 
Theoremssext 3948* An extensionality-like principle that uses the subset instead of the membership relation: two classes are equal iff they have the same subsets. (Contributed by NM, 30-Jun-2004.)
(A = Bx(xAxB))
 
Theoremnssssr 3949* Negation of subclass relationship. Compare nssr 2997. (Contributed by Jim Kingdon, 17-Sep-2018.)
(x(xA ¬ xB) → ¬ AB)
 
Theorempweqb 3950 Classes are equal if and only if their power classes are equal. Exercise 19 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
(A = B ↔ 𝒫 A = 𝒫 B)
 
Theoremintid 3951* The intersection of all sets to which a set belongs is the singleton of that set. (Contributed by NM, 5-Jun-2009.)
A V        {xA x} = {A}
 
Theoremeuabex 3952 The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.)
(∃!xφ → {xφ} V)
 
Theoremmss 3953* An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
(y y Ax(xA z z x))
 
Theoremexss 3954* Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)
(x A φy(yA x y φ))
 
Theoremopexg 3955 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.)
((A 𝑉 B 𝑊) → ⟨A, B V)
 
TheoremopexgOLD 3956 An ordered pair of sets is a set. This is a special case of opexg 3955 and new proofs should use opexg 3955 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3955 and then remove it.
((A V B V) → ⟨A, B V)
 
Theoremopex 3957 An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.)
A V    &   B V       A, B V
 
Theoremotexg 3958 An ordered triple of sets is a set. (Contributed by Jim Kingdon, 19-Sep-2018.)
((A 𝑈 B 𝑉 𝐶 𝑊) → ⟨A, B, 𝐶 V)
 
Theoremelop 3959 An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V    &   𝐶 V       (A B, 𝐶⟩ ↔ (A = {B} A = {B, 𝐶}))
 
Theoremopi1 3960 One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V       {A} A, B
 
Theoremopi2 3961 One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V       {A, B} A, B
 
2.3.3  Ordered pair theorem
 
Theoremopm 3962* An ordered pair is inhabited iff the arguments are sets. (Contributed by Jim Kingdon, 21-Sep-2018.)
(x x A, B⟩ ↔ (A V B V))
 
Theoremopnzi 3963 An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 3962). (Contributed by Mario Carneiro, 26-Apr-2015.)
A V    &   B V       A, B⟩ ≠ ∅
 
Theoremopth1 3964 Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V       (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → A = 𝐶)
 
Theoremopth 3965 The ordered pair theorem. If two ordered pairs are equal, their first elements are equal and their second elements are equal. Exercise 6 of [TakeutiZaring] p. 16. Note that 𝐶 and 𝐷 are not required to be sets due our specific ordered pair definition. (Contributed by NM, 28-May-1995.)
A V    &   B V       (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷))
 
Theoremopthg 3966 Ordered pair theorem. 𝐶 and 𝐷 are not required to be sets under our specific ordered pair definition. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
((A 𝑉 B 𝑊) → (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷)))
 
Theoremopthg2 3967 Ordered pair theorem. (Contributed by NM, 14-Oct-2005.) (Revised by Mario Carneiro, 26-Apr-2015.)
((𝐶 𝑉 𝐷 𝑊) → (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷)))
 
Theoremopth2 3968 Ordered pair theorem. (Contributed by NM, 21-Sep-2014.)
𝐶 V    &   𝐷 V       (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ ↔ (A = 𝐶 B = 𝐷))
 
Theoremotth2 3969 Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V    &   𝑅 V       (⟨⟨A, B⟩, 𝑅⟩ = ⟨⟨𝐶, 𝐷⟩, 𝑆⟩ ↔ (A = 𝐶 B = 𝐷 𝑅 = 𝑆))
 
Theoremotth 3970 Ordered triple theorem. (Contributed by NM, 25-Sep-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V    &   𝑅 V       (⟨A, B, 𝑅⟩ = ⟨𝐶, 𝐷, 𝑆⟩ ↔ (A = 𝐶 B = 𝐷 𝑅 = 𝑆))
 
Theoremeqvinop 3971* A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.)
B V    &   𝐶 V       (A = ⟨B, 𝐶⟩ ↔ xy(A = ⟨x, yx, y⟩ = ⟨B, 𝐶⟩))
 
Theoremcopsexg 3972* Substitution of class A for ordered pair x, y. (Contributed by NM, 27-Dec-1996.) (Revised by Andrew Salmon, 11-Jul-2011.)
(A = ⟨x, y⟩ → (φxy(A = ⟨x, y φ)))
 
Theoremcopsex2t 3973* Closed theorem form of copsex2g 3974. (Contributed by NM, 17-Feb-2013.)
((xy((x = A y = B) → (φψ)) (A 𝑉 B 𝑊)) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
 
Theoremcopsex2g 3974* Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.)
((x = A y = B) → (φψ))       ((A 𝑉 B 𝑊) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
 
Theoremcopsex4g 3975* An implicit substitution inference for 2 ordered pairs. (Contributed by NM, 5-Aug-1995.)
(((x = A y = B) (z = 𝐶 w = 𝐷)) → (φψ))       (((A 𝑅 B 𝑆) (𝐶 𝑅 𝐷 𝑆)) → (xyzw((⟨A, B⟩ = ⟨x, y𝐶, 𝐷⟩ = ⟨z, w⟩) φ) ↔ ψ))
 
Theorem0nelop 3976 A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)
¬ ∅ A, B
 
Theoremopeqex 3977 Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.)
(⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ((A V B V) ↔ (𝐶 V 𝐷 V)))
 
Theoremopcom 3978 An ordered pair commutes iff its members are equal. (Contributed by NM, 28-May-2009.)
A V    &   B V       (⟨A, B⟩ = ⟨B, A⟩ ↔ A = B)
 
Theoremmoop2 3979* "At most one" property of an ordered pair. (Contributed by NM, 11-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
B V       ∃*x A = ⟨B, x
 
Theoremopeqsn 3980 Equivalence for an ordered pair equal to a singleton. (Contributed by NM, 3-Jun-2008.)
A V    &   B V    &   𝐶 V       (⟨A, B⟩ = {𝐶} ↔ (A = B 𝐶 = {A}))
 
Theoremopeqpr 3981 Equivalence for an ordered pair equal to an unordered pair. (Contributed by NM, 3-Jun-2008.)
A V    &   B V    &   𝐶 V    &   𝐷 V       (⟨A, B⟩ = {𝐶, 𝐷} ↔ ((𝐶 = {A} 𝐷 = {A, B}) (𝐶 = {A, B} 𝐷 = {A})))
 
Theoremeuotd 3982* Prove existential uniqueness for an ordered triple. (Contributed by Mario Carneiro, 20-May-2015.)
(φA V)    &   (φB V)    &   (φ𝐶 V)    &   (φ → (ψ ↔ (𝑎 = A 𝑏 = B 𝑐 = 𝐶)))       (φ∃!x𝑎𝑏𝑐(x = ⟨𝑎, 𝑏, 𝑐 ψ))
 
Theoremuniop 3983 The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V       A, B⟩ = {A, B}
 
Theoremuniopel 3984 Ordered pair membership is inherited by class union. (Contributed by NM, 13-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
A V    &   B V       (⟨A, B 𝐶A, B 𝐶)
 
2.3.4  Ordered-pair class abstractions (cont.)
 
Theoremopabid 3985 The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(⟨x, y {⟨x, y⟩ ∣ φ} ↔ φ)
 
Theoremelopab 3986* Membership in a class abstraction of pairs. (Contributed by NM, 24-Mar-1998.)
(A {⟨x, y⟩ ∣ φ} ↔ xy(A = ⟨x, y φ))
 
TheoremopelopabsbALT 3987* The law of concretion in terms of substitutions. Less general than opelopabsb 3988, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
(⟨z, w {⟨x, y⟩ ∣ φ} ↔ [w / y][z / x]φ)
 
Theoremopelopabsb 3988* The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
(⟨A, B {⟨x, y⟩ ∣ φ} ↔ [A / x][B / y]φ)
 
Theorembrabsb 3989* The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
𝑅 = {⟨x, y⟩ ∣ φ}       (A𝑅B[A / x][B / y]φ)
 
Theoremopelopabt 3990* Closed theorem form of opelopab 3999. (Contributed by NM, 19-Feb-2013.)
((xy(x = A → (φψ)) xy(y = B → (ψχ)) (A 𝑉 B 𝑊)) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ))
 
Theoremopelopabga 3991* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
((x = A y = B) → (φψ))       ((A 𝑉 B 𝑊) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ ψ))
 
Theorembrabga 3992* The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
((x = A y = B) → (φψ))    &   𝑅 = {⟨x, y⟩ ∣ φ}       ((A 𝑉 B 𝑊) → (A𝑅Bψ))
 
Theoremopelopab2a 3993* Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
((x = A y = B) → (φψ))       ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ↔ ψ))
 
Theoremopelopaba 3994* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
A V    &   B V    &   ((x = A y = B) → (φψ))       (⟨A, B {⟨x, y⟩ ∣ φ} ↔ ψ)
 
Theorembraba 3995* The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)
A V    &   B V    &   ((x = A y = B) → (φψ))    &   𝑅 = {⟨x, y⟩ ∣ φ}       (A𝑅Bψ)
 
Theoremopelopabg 3996* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
(x = A → (φψ))    &   (y = B → (ψχ))       ((A 𝑉 B 𝑊) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ))
 
Theorembrabg 3997* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.)
(x = A → (φψ))    &   (y = B → (ψχ))    &   𝑅 = {⟨x, y⟩ ∣ φ}       ((A 𝐶 B 𝐷) → (A𝑅Bχ))
 
Theoremopelopab2 3998* Ordered pair membership in an ordered pair class abstraction. (Contributed by NM, 14-Oct-2007.) (Revised by Mario Carneiro, 19-Dec-2013.)
(x = A → (φψ))    &   (y = B → (ψχ))       ((A 𝐶 B 𝐷) → (⟨A, B {⟨x, y⟩ ∣ ((x 𝐶 y 𝐷) φ)} ↔ χ))
 
Theoremopelopab 3999* The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
A V    &   B V    &   (x = A → (φψ))    &   (y = B → (ψχ))       (⟨A, B {⟨x, y⟩ ∣ φ} ↔ χ)
 
Theorembrab 4000* The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)
A V    &   B V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   𝑅 = {⟨x, y⟩ ∣ φ}       (A𝑅Bχ)
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