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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cdeqnot 2901 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal 2902* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab 2903* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal1 2904* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab1 2905* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqim 2906 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqcv 2907 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqeq 2908 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqel 2909 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | nfcdeq 2910* | 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 2911* | Variation of nfcdeq 2910 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | ru 2912 |
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 4054. (Contributed by NM, 7-Aug-1994.) |
Syntax | wsbc 2913 | 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 2914 |
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 2938 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 2915 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 2915, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2914 in the form of sbc8g 2920. 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 2914 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 2915 |
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 2914 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 2916 instead of df-sbc 2914. (dfsbcq2 2916 is needed because
unlike Quine we do not overload the df-sb 1737 syntax.) As a consequence of
these theorems, we can derive sbc8g 2920, which is a weaker version of
df-sbc 2914 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2920, so we will allow direct use of df-sbc 2914. 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 2916 | 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 1737 and substitution for class variables df-sbc 2914. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2915. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbsbc 2917 | Show that df-sb 1737 and df-sbc 2914 are equivalent when the class term in df-sbc 2914 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1737 for proofs involving df-sbc 2914. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Theorem | sbceq1d 2918 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceq1dd 2919 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbc8g 2920 | This is the closest we can get to df-sbc 2914 if we start from dfsbcq 2915 (see its comments) and dfsbcq2 2916. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Theorem | sbcex 2921 | 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 2922 | Equality theorem for class substitution. Class version of sbequ12 1745. (Contributed by NM, 26-Sep-2003.) |
Theorem | sbceq2a 2923 | Equality theorem for class substitution. Class version of sbequ12r 1746. (Contributed by NM, 4-Jan-2017.) |
Theorem | spsbc 2924 | 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 1749 and rspsbc 2995. (Contributed by NM, 16-Jan-2004.) |
Theorem | spsbcd 2925 | 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 1749 and rspsbc 2995. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth 2926 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |
Theorem | sbcthdv 2927* | Deduction version of sbcth 2926. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbcid 2928 | An identity theorem for substitution. See sbid 1748. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Theorem | nfsbc1d 2929 | Deduction version of nfsbc1 2930. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1 2930 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1v 2931* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbcd 2932 | Deduction version of nfsbc 2933. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc 2933 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbcco 2934* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcco2 2935* | 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 2936* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbc6g 2937* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc6 2938* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Theorem | sbc7 2939* | An equivalence for class substitution in the spirit of df-clab 2127. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | cbvsbcw 2940* | Version of cbvsbc 2941 with a disjoint variable condition. (Contributed by Gino Giotto, 10-Jan-2024.) |
Theorem | cbvsbc 2941 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvsbcv 2942* | 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 2943* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2944.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegf 2944* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcieg 2945* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Theorem | sbcie2g 2946* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2947 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | sbcie 2947* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Theorem | sbciedf 2948* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcied 2949* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | sbcied2 2950* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | elrabsf 2951 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2842 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 2952* | Substitution applied to an atomic wff. Set theory version of eqsb3 2244. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcng 2953 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcimg 2954 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcan 2955 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcang 2956 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcor 2957 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcorg 2958 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcbig 2959 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | sbcn1 2960 | Move negation in and out of class substitution. One direction of sbcng 2953 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcim1 2961 | Distribution of class substitution over implication. One direction of sbcimg 2954 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi1 2962 | Distribution of class substitution over biconditional. One direction of sbcbig 2959 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi2 2963 | Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
Theorem | sbcal 2964* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcalg 2965* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcex2 2966* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Theorem | sbcexg 2967* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Theorem | sbceqal 2968* | A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.) |
Theorem | sbeqalb 2969* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |
Theorem | sbcbid 2970 | Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcbidv 2971* | Formula-building deduction for class substitution. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcbii 2972 | Formula-building inference for class substitution. (Contributed by NM, 11-Nov-2005.) |
Theorem | eqsbc3r 2973* | eqsbc3 2952 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 2974 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.) |
Theorem | sbcel1v 2975* | Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcel2gv 2976* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcel21v 2977* | Class substitution into a membership relation. One direction of sbcel2gv 2976 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcimdv 2978* | Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1434). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
Theorem | sbctt 2979 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Theorem | sbcgf 2980 | 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 2981 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |
Theorem | sbcg 2982* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2980. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | sbc2iegf 2983* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Theorem | sbc2ie 2984* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Theorem | sbc2iedv 2985* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
Theorem | sbc3ie 2986* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |
Theorem | sbccomlem 2987* | Lemma for sbccom 2988. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |
Theorem | sbccom 2988* | Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
Theorem | sbcralt 2989* | Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) |
Theorem | sbcrext 2990* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcralg 2991* | Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcrex 2992* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
Theorem | sbcreug 2993* | Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.) |
Theorem | sbcabel 2994* | Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.) |
Theorem | rspsbc 2995* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. This provides an axiom for a predicate calculus for a restricted domain. This theorem generalizes the unrestricted stdpc4 1749 and spsbc 2924. See also rspsbca 2996 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | rspsbca 2996* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.) |
Theorem | rspesbca 2997* | Existence form of rspsbca 2996. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | spesbc 2998 | Existence form of spsbc 2924. (Contributed by Mario Carneiro, 18-Nov-2016.) |
Theorem | spesbcd 2999 | form of spsbc 2924. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth2 3000* | A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
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