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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cdeqi 2801 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqri 2802 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqth 2803 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqnot 2804 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal 2805* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab 2806* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal1 2807* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab1 2808* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqim 2809 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqcv 2810 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqeq 2811 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqel 2812 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | nfcdeq 2813* | 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 2814* | Variation of nfcdeq 2813 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | ru 2815 |
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 3904. (Contributed by NM, 7-Aug-1994.) |
Syntax | wsbc 2816 | 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 2817 |
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 2841 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 2818 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 2818, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2817 in the form of sbc8g 2823. 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 2817 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 2818 |
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 2817 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 2819 instead of df-sbc 2817. (dfsbcq2 2819 is needed because
unlike Quine we do not overload the df-sb 1687 syntax.) As a consequence of
these theorems, we can derive sbc8g 2823, which is a weaker version of
df-sbc 2817 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2823, so we will allow direct use of df-sbc 2817. 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 2819 | 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 1687 and substitution for class variables df-sbc 2817. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2818. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbsbc 2820 | Show that df-sb 1687 and df-sbc 2817 are equivalent when the class term in df-sbc 2817 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1687 for proofs involving df-sbc 2817. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Theorem | sbceq1d 2821 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceq1dd 2822 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbc8g 2823 | This is the closest we can get to df-sbc 2817 if we start from dfsbcq 2818 (see its comments) and dfsbcq2 2819. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Theorem | sbcex 2824 | 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 2825 | Equality theorem for class substitution. Class version of sbequ12 1695. (Contributed by NM, 26-Sep-2003.) |
Theorem | sbceq2a 2826 | Equality theorem for class substitution. Class version of sbequ12r 1696. (Contributed by NM, 4-Jan-2017.) |
Theorem | spsbc 2827 | 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 1699 and rspsbc 2897. (Contributed by NM, 16-Jan-2004.) |
Theorem | spsbcd 2828 | 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 1699 and rspsbc 2897. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth 2829 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |
Theorem | sbcthdv 2830* | Deduction version of sbcth 2829. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbcid 2831 | An identity theorem for substitution. See sbid 1698. (Contributed by Mario Carneiro, 18-Feb-2017.) |
Theorem | nfsbc1d 2832 | Deduction version of nfsbc1 2833. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1 2833 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc1v 2834* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbcd 2835 | Deduction version of nfsbc 2836. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | nfsbc 2836 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbcco 2837* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcco2 2838* | 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 2839* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | sbc6g 2840* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | sbc6 2841* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |
Theorem | sbc7 2842* | An equivalence for class substitution in the spirit of df-clab 2069. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | cbvsbc 2843 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvsbcv 2844* | 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 2845* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2846.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbciegf 2846* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcieg 2847* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |
Theorem | sbcie2g 2848* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2849 avoids a disjointness condition on and by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | sbcie 2849* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |
Theorem | sbciedf 2850* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcied 2851* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | sbcied2 2852* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |
Theorem | elrabsf 2853 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2748 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 2854* | Substitution applied to an atomic wff. Set theory version of eqsb3 2183. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcng 2855 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcimg 2856 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcan 2857 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcang 2858 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcor 2859 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcorg 2860 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |
Theorem | sbcbig 2861 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | sbcn1 2862 | Move negation in and out of class substitution. One direction of sbcng 2855 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcim1 2863 | Distribution of class substitution over implication. One direction of sbcimg 2856 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi1 2864 | Distribution of class substitution over biconditional. One direction of sbcbig 2861 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcbi2 2865 | Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) |
Theorem | sbcal 2866* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbcalg 2867* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |
Theorem | sbcex2 2868* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Theorem | sbcexg 2869* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |
Theorem | sbceqal 2870* | A variation of extensionality for classes. (Contributed by Andrew Salmon, 28-Jun-2011.) |
Theorem | sbeqalb 2871* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |
Theorem | sbcbid 2872 | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcbidv 2873* | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |
Theorem | sbcbii 2874 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) |
Theorem | eqsbc3r 2875* | eqsbc3 2854 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 2876 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Revised by NM, 17-Aug-2018.) |
Theorem | sbcel1v 2877* | Class substitution into a membership relation. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcel2gv 2878* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcel21v 2879* | Class substitution into a membership relation. One direction of sbcel2gv 2878 that holds for proper classes. (Contributed by NM, 17-Aug-2018.) |
Theorem | sbcimdv 2880* | Substitution analogue of Theorem 19.20 of [Margaris] p. 90 (alim 1387). (Contributed by NM, 11-Nov-2005.) (Revised by NM, 17-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.) |
Theorem | sbctt 2881 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Theorem | sbcgf 2882 | 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 2883 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |
Theorem | sbcg 2884* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2882. (Contributed by Alan Sare, 10-Nov-2012.) |
Theorem | sbc2iegf 2885* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Theorem | sbc2ie 2886* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Theorem | sbc2iedv 2887* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
Theorem | sbc3ie 2888* | Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.) |
Theorem | sbccomlem 2889* | Lemma for sbccom 2890. (Contributed by NM, 14-Nov-2005.) (Revised by Mario Carneiro, 18-Oct-2016.) |
Theorem | sbccom 2890* | Commutative law for double class substitution. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Mario Carneiro, 18-Oct-2016.) |
Theorem | sbcralt 2891* | Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.) |
Theorem | sbcrext 2892* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | sbcralg 2893* | Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Theorem | sbcrex 2894* | Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.) |
Theorem | sbcreug 2895* | Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.) |
Theorem | sbcabel 2896* | Interchange class substitution and class abstraction. (Contributed by NM, 5-Nov-2005.) |
Theorem | rspsbc 2897* | 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 1699 and spsbc 2827. See also rspsbca 2898 and rspcsbela . (Contributed by NM, 17-Nov-2006.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | rspsbca 2898* | Restricted quantifier version of Axiom 4 of [Mendelson] p. 69. (Contributed by NM, 14-Dec-2005.) |
Theorem | rspesbca 2899* | Existence form of rspsbca 2898. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |
Theorem | spesbc 2900 | Existence form of spsbc 2827. (Contributed by Mario Carneiro, 18-Nov-2016.) |
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