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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mob2 2901* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |
Theorem | moi2 2902* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |
Theorem | mob 2903* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | moi 2904* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | morex 2905* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | euxfr2dc 2906* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
DECID | ||
Theorem | euxfrdc 2907* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
DECID | ||
Theorem | euind 2908* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
Theorem | reu2 2909* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
Theorem | reu6 2910* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
Theorem | reu3 2911* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu6i 2912* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | eqreu 2913* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | rmo4 2914* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | reu4 2915* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
Theorem | reu7 2916* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu8 2917* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | rmo3f 2918* | Restricted "at most one" using explicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
Theorem | rmo4f 2919* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
Theorem | reueq 2920* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Theorem | rmoan 2921 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
Theorem | rmoim 2922 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimia 2923 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimi2 2924 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reuswapdc 2925* | 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 2926* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
Theorem | 2rmorex 2927* | Double restricted quantification with "at most one," analogous to 2moex 2099. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | nelrdva 2928* | 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 ternary 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 condition: 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 2933 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 disjoint variable conditions of cdeqab1 2938 and cdeqab 2936. | ||
Syntax | wcdeq 2929 | 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 2930 | 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 2931 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqri 2932 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqth 2933 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqnot 2934 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal 2935* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab 2936* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal1 2937* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab1 2938* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqim 2939 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqcv 2940 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqeq 2941 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqel 2942 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | nfcdeq 2943* | 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 2944* | Variation of nfcdeq 2943 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | ru 2945 |
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 4094. (Contributed by NM, 7-Aug-1994.) |
Syntax | wsbc 2946 | 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 2947 |
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 2971 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 2948 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 2948, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2947 in the form of sbc8g 2953. 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 2947 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 2948 |
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 2947 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 2949 instead of df-sbc 2947. (dfsbcq2 2949 is needed because
unlike Quine we do not overload the df-sb 1750 syntax.) As a consequence of
these theorems, we can derive sbc8g 2953, which is a weaker version of
df-sbc 2947 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2953, so we will allow direct use of df-sbc 2947. 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 2949 | 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 1750 and substitution for class variables df-sbc 2947. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2948. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbsbc 2950 | Show that df-sb 1750 and df-sbc 2947 are equivalent when the class term in df-sbc 2947 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1750 for proofs involving df-sbc 2947. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Theorem | sbceq1d 2951 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceq1dd 2952 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbc8g 2953 | This is the closest we can get to df-sbc 2947 if we start from dfsbcq 2948 (see its comments) and dfsbcq2 2949. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Theorem | sbcex 2954 | 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 2955 | Equality theorem for class substitution. Class version of sbequ12 1758. (Contributed by NM, 26-Sep-2003.) |
Theorem | sbceq2a 2956 | Equality theorem for class substitution. Class version of sbequ12r 1759. (Contributed by NM, 4-Jan-2017.) |
Theorem | spsbc 2957 | 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 1762 and rspsbc 3028. (Contributed by NM, 16-Jan-2004.) |
Theorem | spsbcd 2958 | 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 1762 and rspsbc 3028. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth 2959 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |
Theorem | sbcthdv 2960* | Deduction version of sbcth 2959. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbcid 2961 | An identity theorem for substitution. See sbid 1761. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Theorem | nfsbc1d 2962 | Deduction version of nfsbc1 2963. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1 2963 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1v 2964* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbcd 2965 | Deduction version of nfsbc 2966. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc 2966 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbcco 2967* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcco2 2968* | 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 2969* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbc6g 2970* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc6 2971* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Theorem | sbc7 2972* | An equivalence for class substitution in the spirit of df-clab 2151. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | cbvsbcw 2973* | Version of cbvsbc 2974 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
Theorem | cbvsbc 2974 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvsbcv 2975* | 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 2976* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2977.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegf 2977* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcieg 2978* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Theorem | sbcie2g 2979* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2980 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | sbcie 2980* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Theorem | sbciedf 2981* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcied 2982* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | sbcied2 2983* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | elrabsf 2984 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2875 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 2985* | Substitution applied to an atomic wff. Set theory version of eqsb3 2268. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcng 2986 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcimg 2987 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcan 2988 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcang 2989 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcor 2990 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcorg 2991 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcbig 2992 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | sbcn1 2993 | Move negation in and out of class substitution. One direction of sbcng 2986 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcim1 2994 | Distribution of class substitution over implication. One direction of sbcimg 2987 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi1 2995 | Distribution of class substitution over biconditional. One direction of sbcbig 2992 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi2 2996 | Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
Theorem | sbcal 2997* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcalg 2998* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcex2 2999* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Theorem | sbcexg 3000* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
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