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Type | Label | Description |
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Statement | ||
Theorem | reu7 2801* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu8 2802* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reueq 2803* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Theorem | rmoan 2804 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
Theorem | rmoim 2805 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimia 2806 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimi2 2807 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reuswapdc 2808* | 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 2809* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
Theorem | 2rmorex 2810* | Double restricted quantification with "at most one," analogous to 2moex 2031. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | nelrdva 2811* | 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 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 2816 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 2821 and cdeqab 2819. | ||
Syntax | wcdeq 2812 | 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 2813 | 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 2814 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqri 2815 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqth 2816 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqnot 2817 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal 2818* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab 2819* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal1 2820* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab1 2821* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqim 2822 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqcv 2823 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqeq 2824 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqel 2825 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | nfcdeq 2826* | 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 2827* | Variation of nfcdeq 2826 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | ru 2828 |
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 3932. (Contributed by NM, 7-Aug-1994.) |
Syntax | wsbc 2829 | 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 2830 |
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 2854 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 2831 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 2831, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2830 in the form of sbc8g 2836. 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 2830 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 2831 |
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 2830 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 2832 instead of df-sbc 2830. (dfsbcq2 2832 is needed because
unlike Quine we do not overload the df-sb 1690 syntax.) As a consequence of
these theorems, we can derive sbc8g 2836, which is a weaker version of
df-sbc 2830 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2836, so we will allow direct use of df-sbc 2830. 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 2832 | 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 1690 and substitution for class variables df-sbc 2830. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2831. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbsbc 2833 | Show that df-sb 1690 and df-sbc 2830 are equivalent when the class term in df-sbc 2830 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1690 for proofs involving df-sbc 2830. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Theorem | sbceq1d 2834 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceq1dd 2835 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbc8g 2836 | This is the closest we can get to df-sbc 2830 if we start from dfsbcq 2831 (see its comments) and dfsbcq2 2832. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Theorem | sbcex 2837 | 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 2838 | Equality theorem for class substitution. Class version of sbequ12 1698. (Contributed by NM, 26-Sep-2003.) |
Theorem | sbceq2a 2839 | Equality theorem for class substitution. Class version of sbequ12r 1699. (Contributed by NM, 4-Jan-2017.) |
Theorem | spsbc 2840 | 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 1702 and rspsbc 2910. (Contributed by NM, 16-Jan-2004.) |
Theorem | spsbcd 2841 | 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 1702 and rspsbc 2910. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth 2842 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |
Theorem | sbcthdv 2843* | Deduction version of sbcth 2842. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbcid 2844 | An identity theorem for substitution. See sbid 1701. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Theorem | nfsbc1d 2845 | Deduction version of nfsbc1 2846. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1 2846 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1v 2847* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbcd 2848 | Deduction version of nfsbc 2849. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc 2849 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbcco 2850* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcco2 2851* | 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 2852* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbc6g 2853* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc6 2854* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Theorem | sbc7 2855* | An equivalence for class substitution in the spirit of df-clab 2072. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | cbvsbc 2856 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvsbcv 2857* | 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 2858* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2859.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegf 2859* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcieg 2860* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Theorem | sbcie2g 2861* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2862 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | sbcie 2862* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Theorem | sbciedf 2863* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcied 2864* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | sbcied2 2865* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | elrabsf 2866 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2760 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 2867* | Substitution applied to an atomic wff. Set theory version of eqsb3 2188. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcng 2868 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcimg 2869 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcan 2870 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcang 2871 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcor 2872 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcorg 2873 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcbig 2874 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | sbcn1 2875 | Move negation in and out of class substitution. One direction of sbcng 2868 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcim1 2876 | Distribution of class substitution over implication. One direction of sbcimg 2869 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi1 2877 | Distribution of class substitution over biconditional. One direction of sbcbig 2874 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi2 2878 | Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
Theorem | sbcal 2879* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcalg 2880* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcex2 2881* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Theorem | sbcexg 2882* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Theorem | sbceqal 2883* | A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.) |
Theorem | sbeqalb 2884* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |
Theorem | sbcbid 2885 | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcbidv 2886* | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcbii 2887 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) |
Theorem | eqsbc3r 2888* | eqsbc3 2867 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.) |
Theorem | sbc3an 2889 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.) |
Theorem | sbcel1v 2890* | Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcel2gv 2891* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcel21v 2892* | Class substitution into a membership relation. One direction of sbcel2gv 2891 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcimdv 2893* | Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1389). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
Theorem | sbctt 2894 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Theorem | sbcgf 2895 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbc19.21g 2896 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |
Theorem | sbcg 2897* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2895. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | sbc2iegf 2898* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Theorem | sbc2ie 2899* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Theorem | sbc2iedv 2900* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
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