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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rexab2 2701* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Theorem | rexrab2 2702* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Theorem | abidnf 2703* | 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.) |
Theorem | dedhb 2704* | A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1431 and nfab 2179 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 2703 is useful. (Contributed by NM, 8-Dec-2006.) |
Theorem | eqeu 2705* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
Theorem | eueq 2706* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |
Theorem | eueq1 2707* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Theorem | eueq2dc 2708* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
DECID | ||
Theorem | eueq3dc 2709* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
DECID DECID | ||
Theorem | moeq 2710* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |
Theorem | moeq3dc 2711* | "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.) |
DECID DECID | ||
Theorem | mosubt 2712* | "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.) |
Theorem | mosub 2713* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
Theorem | mo2icl 2714* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |
Theorem | mob2 2715* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |
Theorem | moi2 2716* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |
Theorem | mob 2717* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | moi 2718* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | morex 2719* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | euxfr2dc 2720* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
DECID | ||
Theorem | euxfrdc 2721* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
DECID | ||
Theorem | euind 2722* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
Theorem | reu2 2723* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
Theorem | reu6 2724* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
Theorem | reu3 2725* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu6i 2726* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | eqreu 2727* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | rmo4 2728* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | reu4 2729* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
Theorem | reu7 2730* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu8 2731* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reueq 2732* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Theorem | rmoan 2733 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
Theorem | rmoim 2734 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimia 2735 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimi2 2736 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reuswapdc 2737* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
DECID | ||
Theorem | reuind 2738* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
Theorem | 2rmorex 2739* | Double restricted quantification with "at most one," analogous to 2moex 1983. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | nelrdva 2740* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
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 2745 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 2750 and cdeqab 2748. | ||
Syntax | wcdeq 2741 | 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 | ||
Definition | df-cdeq 2742 | 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 | ||
Theorem | cdeqi 2743 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqri 2744 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqth 2745 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqnot 2746 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal 2747* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab 2748* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal1 2749* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab1 2750* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqim 2751 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqcv 2752 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqeq 2753 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqel 2754 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | nfcdeq 2755* | 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 | ||
Theorem | nfccdeq 2756* | Variation of nfcdeq 2755 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | ru 2757 |
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 3866. (Contributed by NM, 7-Aug-1994.) |
Syntax | wsbc 2758 | 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 ." |
Definition | df-sbc 2759 |
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 2783 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 2760 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 2760, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2759 in the form of sbc8g 2765. 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 2759 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.) |
Theorem | dfsbcq 2760 |
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 2759 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 2761 instead of df-sbc 2759. (dfsbcq2 2761 is needed because
unlike Quine we do not overload the df-sb 1643 syntax.) As a consequence of
these theorems, we can derive sbc8g 2765, which is a weaker version of
df-sbc 2759 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2765, so we will allow direct use of df-sbc 2759. 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.) |
Theorem | dfsbcq2 2761 | 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 1643 and substitution for class variables df-sbc 2759. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2760. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbsbc 2762 | Show that df-sb 1643 and df-sbc 2759 are equivalent when the class term in df-sbc 2759 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1643 for proofs involving df-sbc 2759. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Theorem | sbceq1d 2763 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceq1dd 2764 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbc8g 2765 | This is the closest we can get to df-sbc 2759 if we start from dfsbcq 2760 (see its comments) and dfsbcq2 2761. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Theorem | sbcex 2766 | 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.) |
Theorem | sbceq1a 2767 | Equality theorem for class substitution. Class version of sbequ12 1651. (Contributed by NM, 26-Sep-2003.) |
Theorem | sbceq2a 2768 | Equality theorem for class substitution. Class version of sbequ12r 1652. (Contributed by NM, 4-Jan-2017.) |
Theorem | spsbc 2769 | 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 1655 and rspsbc 2834. (Contributed by NM, 16-Jan-2004.) |
Theorem | spsbcd 2770 | 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 1655 and rspsbc 2834. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth 2771 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |
Theorem | sbcthdv 2772* | Deduction version of sbcth 2771. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbcid 2773 | An identity theorem for substitution. See sbid 1654. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Theorem | nfsbc1d 2774 | Deduction version of nfsbc1 2775. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1 2775 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1v 2776* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbcd 2777 | Deduction version of nfsbc 2778. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc 2778 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbcco 2779* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcco2 2780* | A composition law for class substitution. Importantly, may occur free in the class expression substituted for . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc5 2781* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbc6g 2782* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc6 2783* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Theorem | sbc7 2784* | An equivalence for class substitution in the spirit of df-clab 2024. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | cbvsbc 2785 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvsbcv 2786* | Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegft 2787* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2788.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegf 2788* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcieg 2789* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Theorem | sbcie2g 2790* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2791 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | sbcie 2791* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Theorem | sbciedf 2792* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcied 2793* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | sbcied2 2794* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | elrabsf 2795 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2690 has implicit substitution). The hypothesis specifies that must not be a free variable in . (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | eqsbc3 2796* | Substitution applied to an atomic wff. Set theory version of eqsb3 2138. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcng 2797 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcimg 2798 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcan 2799 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcang 2800 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
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