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Theorem List for Intuitionistic Logic Explorer - 2701-2800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclel3g 2701* An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)

Theoremclel3 2702* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theoremclel4 2703* An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Theorempm13.183 2704* Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.)

Theoremrr19.3v 2705* Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)

Theoremrr19.28v 2706* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)

Theoremelabgt 2707* Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2711.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremelabgf 2708 Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremelabf 2709* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremelab 2710* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)

Theoremelabg 2711* Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)

Theoremelab2g 2712* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab2 2713* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)

Theoremelab4g 2714* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.)

Theoremelab3gf 2715 Membership in a class abstraction, with a weaker antecedent than elabgf 2708. (Contributed by NM, 6-Sep-2011.)

Theoremelab3g 2716* Membership in a class abstraction, with a weaker antecedent than elabg 2711. (Contributed by NM, 29-Aug-2006.)

Theoremelab3 2717* Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.)

Theoremelrabi 2718* Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.)

Theoremelrabf 2719 Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.)

Theoremelrab3t 2720* Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2722.) (Contributed by Thierry Arnoux, 31-Aug-2017.)

Theoremelrab 2721* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)

Theoremelrab3 2722* Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)

Theoremelrab2 2723* Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.)

Theoremralab 2724* Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremralrab 2725* Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)

Theoremrexab 2726* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremrexrab 2727* Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)

Theoremralab2 2728* Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremralrab2 2729* Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremrexab2 2730* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremrexrab2 2731* Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.)

Theoremabidnf 2732* Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.)

Theoremdedhb 2733* A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1450 and nfab 2198 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 2732 is useful. (Contributed by NM, 8-Dec-2006.)

Theoremeqeu 2734* A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.)

Theoremeueq 2735* Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.)

Theoremeueq1 2736* Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.)

Theoremeueq2dc 2737* Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.)
DECID

Theoremeueq3dc 2738* Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
DECID DECID

Theoremmoeq 2739* There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)

Theoremmoeq3dc 2740* "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)
DECID DECID

Theoremmosubt 2741* "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.)

Theoremmosub 2742* "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.)

Theoremmo2icl 2743* Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)

Theoremmob2 2744* Consequence of "at most one." (Contributed by NM, 2-Jan-2015.)

Theoremmoi2 2745* Consequence of "at most one." (Contributed by NM, 29-Jun-2008.)

Theoremmob 2746* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)

Theoremmoi 2747* Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)

Theoremmorex 2748* Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremeuxfr2dc 2749* Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.)
DECID

Theoremeuxfrdc 2750* Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.)
DECID

Theoremeuind 2751* Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.)

Theoremreu2 2752* A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)

Theoremreu6 2753* A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.)

Theoremreu3 2754* A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)

Theoremreu6i 2755* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremeqreu 2756* A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)

Theoremrmo4 2757* Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.)

Theoremreu4 2758* Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.)

Theoremreu7 2759* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Theoremreu8 2760* Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.)

Theoremreueq 2761* Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.)

Theoremrmoan 2762 Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)

Theoremrmoim 2763 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremrmoimia 2764 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremrmoimi2 2765 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theorem2reuswapdc 2766* A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
DECID

Theoremreuind 2767* Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.)

Theorem2rmorex 2768* Double restricted quantification with "at most one," analogous to 2moex 2002. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremnelrdva 2769* Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)

2.1.7  Conditional equality (experimental)

This is a very useless definition, which "abbreviates" as CondEq . What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific 3-ary operation .

This is all used as part of a metatheorem: we want to say that and are provable, for any expressions or in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations.

The metatheorem comes with a disjoint variables assumption: every variable in is assumed disjoint from except itself. For such a proof by induction, we must consider each of the possible forms of . If it is a variable other than , then we have CondEq or CondEq , which is provable by cdeqth 2774 and reflexivity. Since we are only working with class and wff expressions, it can't be itself in set.mm, but if it was we'd have to also prove CondEq (where set equality is being used on the right).

Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder).

In each of the primitive proofs, we are allowed to assume that is disjoint from and vice versa, because this is maintained through the induction. This is how we satisfy the DV assumptions of cdeqab1 2779 and cdeqab 2777.

Syntaxwcdeq 2770 Extend wff notation to include conditional equality. This is a technical device used in the proof that is the not-free predicate, and that definitions are conservative as a result.
CondEq

Definitiondf-cdeq 2771 Define conditional equality. All the notation to the left of the is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqi 2772 Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqri 2773 Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqth 2774 Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqnot 2775 Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqal 2776* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqab 2777* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqal1 2778* Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqab1 2779* Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq

Theoremcdeqim 2780 Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremcdeqcv 2781 Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremcdeqeq 2782 Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremcdeqel 2783 Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq        CondEq        CondEq

Theoremnfcdeq 2784* If we have a conditional equality proof, where is and is , and in fact does not have free in it according to , then unconditionally. This proves that is actually a not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremnfccdeq 2785* Variation of nfcdeq 2784 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.)
CondEq

Theoremru 2786 Russell's Paradox. Proposition 4.14 of [TakeutiZaring] p. 14.

In the late 1800s, Frege's Axiom of (unrestricted) Comprehension, expressed in our notation as , asserted that any collection of sets is a set i.e. belongs to the universe of all sets. In particular, by substituting (the "Russell class") for , it asserted , meaning that the "collection of all sets which are not members of themselves" is a set. However, here we prove . This contradiction was discovered by Russell in 1901 (published in 1903), invalidating the Comprehension Axiom and leading to the collapse of Frege's system.

In 1908, Zermelo rectified this fatal flaw by replacing Comprehension with a weaker Subset (or Separation) Axiom asserting that is a set only when it is smaller than some other set . The intuitionistic set theory IZF includes such a separation axiom, Axiom 6 of [Crosilla] p. "Axioms of CZF and IZF", which we include as ax-sep 3903. (Contributed by NM, 7-Aug-1994.)

2.1.9  Proper substitution of classes for sets

Syntaxwsbc 2787 Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class for setvar variable in wff ."

Definitiondf-sbc 2788 Define the proper substitution of a class for a set.

When is a proper class, our definition evaluates to false. This is somewhat arbitrary: we could have, instead, chosen the conclusion of sbc6 2812 for our definition, which always evaluates to true for proper classes.

Our definition also does not produce the same results as discussed in the proof of Theorem 6.6 of [Quine] p. 42 (although Theorem 6.6 itself does hold, as shown by dfsbcq 2789 below). Unfortunately, Quine's definition requires a recursive syntactical breakdown of , and it does not seem possible to express it with a single closed formula.

If we did not want to commit to any specific proper class behavior, we could use this definition only to prove theorem dfsbcq 2789, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2788 in the form of sbc8g 2794. However, the behavior of Quine's definition at proper classes is similarly arbitrary, and for practical reasons (to avoid having to prove sethood of in every use of this definition) we allow direct reference to df-sbc 2788 and assert that is always false when is a proper class.

The related definition df-csb defines proper substitution into a class variable (as opposed to a wff variable). (Contributed by NM, 14-Apr-1995.) (Revised by NM, 25-Dec-2016.)

Theoremdfsbcq 2789 This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, provides us with a weak definition of the proper substitution of a class for a set. Since our df-sbc 2788 does not result in the same behavior as Quine's for proper classes, if we wished to avoid conflict with Quine's definition we could start with this theorem and dfsbcq2 2790 instead of df-sbc 2788. (dfsbcq2 2790 is needed because unlike Quine we do not overload the df-sb 1662 syntax.) As a consequence of these theorems, we can derive sbc8g 2794, which is a weaker version of df-sbc 2788 that leaves substitution undefined when is a proper class.

However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2794, so we will allow direct use of df-sbc 2788. Proper substiution with a proper class is rarely needed, and when it is, we can simply use the expansion of Quine's definition. (Contributed by NM, 14-Apr-1995.)

Theoremdfsbcq2 2790 This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1662 and substitution for class variables df-sbc 2788. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2789. (Contributed by NM, 31-Dec-2016.)

Theoremsbsbc 2791 Show that df-sb 1662 and df-sbc 2788 are equivalent when the class term in df-sbc 2788 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1662 for proofs involving df-sbc 2788. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.)

Theoremsbceq1d 2792 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)

Theoremsbceq1dd 2793 Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.)

Theoremsbc8g 2794 This is the closest we can get to df-sbc 2788 if we start from dfsbcq 2789 (see its comments) and dfsbcq2 2790. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)

Theoremsbcex 2795 By our definition of proper substitution, it can only be true if the substituted expression is a set. (Contributed by Mario Carneiro, 13-Oct-2016.)

Theoremsbceq1a 2796 Equality theorem for class substitution. Class version of sbequ12 1670. (Contributed by NM, 26-Sep-2003.)

Theoremsbceq2a 2797 Equality theorem for class substitution. Class version of sbequ12r 1671. (Contributed by NM, 4-Jan-2017.)

Theoremspsbc 2798 Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1674 and rspsbc 2868. (Contributed by NM, 16-Jan-2004.)

Theoremspsbcd 2799 Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1674 and rspsbc 2868. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremsbcth 2800 A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.)

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