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Theorem List for New Foundations Explorer - 2801-2900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrmobiia 2801 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
   =>   
 
Theoremrmobii 2802 Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.)
   =>   
 
Theoremraleqf 2803 Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
 F/_   &     F/_   =>   
 
Theoremrexeqf 2804 Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)
 F/_   &     F/_   =>   
 
Theoremreueq1f 2805 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 5-Apr-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
 F/_   &     F/_   =>   
 
Theoremrmoeq1f 2806 Equality theorem for restricted uniqueness quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 F/_   &     F/_   =>   
 
Theoremraleq 2807* Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
 
Theoremrexeq 2808* Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)
 
Theoremreueq1 2809* Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
 
Theoremrmoeq1 2810* Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 
Theoremraleqi 2811* Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
   =>   
 
Theoremrexeqi 2812* Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)
   =>   
 
Theoremraleqdv 2813* Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.)
   =>   
 
Theoremrexeqdv 2814* Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.)
   =>   
 
Theoremraleqbi1dv 2815* Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
   =>   
 
Theoremrexeqbi1dv 2816* Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)
   =>   
 
Theoremreueqd 2817* Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.)
   =>   
 
Theoremrmoeqd 2818* Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
   =>   
 
Theoremraleqbidv 2819* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
   &       =>   
 
Theoremrexeqbidv 2820* Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.)
   &       =>   
 
Theoremraleqbidva 2821* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
   &       =>   
 
Theoremrexeqbidva 2822* Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.)
   &       =>   
 
Theoremmormo 2823 Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.)
 
Theoremreu5 2824 Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.)
 
Theoremreurex 2825 Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
 
Theoremreurmo 2826 Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.)
 
Theoremrmo5 2827 Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.)
 
Theoremnrexrmo 2828 Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
 
Theoremcbvralf 2829 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
 F/_   &     F/_   &     F/   &     F/   &       =>   
 
Theoremcbvrexf 2830 Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.)
 F/_   &     F/_   &     F/   &     F/   &       =>   
 
Theoremcbvral 2831* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)

 F/   &     F/   &       =>   
 
Theoremcbvrex 2832* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

 F/   &     F/   &       =>   
 
Theoremcbvreu 2833* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)

 F/   &     F/   &       =>   
 
Theoremcbvrmo 2834* Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.)

 F/   &     F/   &       =>   
 
Theoremcbvralv 2835* Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)
   =>   
 
Theoremcbvrexv 2836* Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)
   =>   
 
Theoremcbvreuv 2837* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
   =>   
 
Theoremcbvrmov 2838* Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
   =>   
 
Theoremcbvraldva2 2839* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
   &       =>   
 
Theoremcbvrexdva2 2840* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
   &       =>   
 
Theoremcbvraldva 2841* Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
   =>   
 
Theoremcbvrexdva 2842* Rule used to change the bound variable in a restricted existential quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
   =>   
 
Theoremcbvral2v 2843* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.)
   &       =>   
 
Theoremcbvrex2v 2844* Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.)
   &       =>   
 
Theoremcbvral3v 2845* Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.)
   &       &       =>   
 
Theoremcbvralsv 2846* Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
 
Theoremcbvrexsv 2847* Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
 
Theoremsbralie 2848* Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)
   =>   
 
Theoremrabbiia 2849 Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.)
   =>   
 
Theoremrabbidva 2850* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.)
   =>   
 
Theoremrabbidv 2851* Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.)
   =>   
 
Theoremrabeqf 2852 Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.)
 F/_   &     F/_   =>   
 
Theoremrabeq 2853* Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.)
 
Theoremrabeqbidv 2854* Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
   &       =>   
 
Theoremrabeqbidva 2855* Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
   &       =>   
 
Theoremrabeq2i 2856 Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
   =>   
 
Theoremcbvrab 2857 Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)
 F/_   &     F/_   &     F/   &     F/   &       =>   
 
Theoremcbvrabv 2858* Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.)
   =>   
 
2.1.6  The universal class
 
Syntaxcvv 2859 Extend class notation to include the universal class symbol.
 
Theoremvjust 2860 Soundness justification theorem for df-v 2861. (Contributed by Rodolfo Medina, 27-Apr-2010.)
 
Definitiondf-v 2861 Define the universal class. Definition 5.20 of [TakeutiZaring] p. 21. Also Definition 2.9 of [Quine] p. 19. (Contributed by NM, 5-Aug-1993.)
 
Theoremvex 2862 All setvar variables are sets (see isset 2863). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)
 
Theoremisset 2863* Two ways to say " is a set": A class is a member of the universal class (see df-v 2861) if and only if the class exists (i.e. there exists some set equal to class ). Theorem 6.9 of [Quine] p. 43. Notational convention: We will use the notational device " " to mean " is a set" very frequently, for example in uniex 4317. Note the when is not a set, it is called a proper class. In some theorems, such as uniexg 4316, in order to shorten certain proofs we use the more general antecedent instead of to mean " is a set."

Note that a constant is implicitly considered distinct from all variables. This is why is not included in the distinct variable list, even though df-clel 2349 requires that the expression substituted for not contain . (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 5-Aug-1993.)

 
Theoremissetf 2864 A version of isset that does not require x and A to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
 F/_   =>   
 
Theoremisseti 2865* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theoremissetri 2866* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)
   =>   
 
Theoremelex 2867 If a class is a member of another class, it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 
Theoremelexi 2868 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)
   =>   
 
Theoremelisset 2869* An element of a class exists. (Contributed by NM, 1-May-1995.)
 
Theoremelex22 2870* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
 
Theoremelex2 2871* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
 
Theoremralv 2872 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 
Theoremrexv 2873 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)
 
Theoremreuv 2874 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)
 
Theoremrmov 2875 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 
Theoremrabab 2876 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 
Theoremralcom4 2877* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 
Theoremrexcom4 2878* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
 
Theoremrexcom4a 2879* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
 
Theoremrexcom4b 2880* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
   =>   
 
Theoremceqsalt 2881* Closed theorem version of ceqsalg 2883. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 F/
 
Theoremceqsralt 2882* Restricted quantifier version of ceqsalt 2881. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
 F/
 
Theoremceqsalg 2883* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

 F/   &       =>   
 
Theoremceqsal 2884* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

 F/   &       &       =>   
 
Theoremceqsalv 2885* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)
   &       =>   
 
Theoremceqsralv 2886* Restricted quantifier version of ceqsalv 2885. (Contributed by NM, 21-Jun-2013.)
   =>   
 
Theoremgencl 2887* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
   &       &       =>   
 
Theorem2gencl 2888* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
   &       &       &       &       =>   
 
Theorem3gencl 2889* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)
   &       &       &       &       &       &       =>   
 
Theoremcgsexg 2890* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)
   &       =>   
 
Theoremcgsex2g 2891* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
   &       =>   
 
Theoremcgsex4g 2892* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)
   &       =>   
 
Theoremceqsex 2893* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

 F/   &       &       =>   
 
Theoremceqsexv 2894* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)
   &       =>   
 
Theoremceqsex2 2895* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

 F/   &     F/   &       &       &       &       =>   
 
Theoremceqsex2v 2896* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)
   &       &       &       =>   
 
Theoremceqsex3v 2897* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)
   &       &       &       &       &       =>   
 
Theoremceqsex4v 2898* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
   &       &       &       &       &       &       &       =>   
 
Theoremceqsex6v 2899* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
   &       &       &       &       &       &       &       &       &       &       &       =>   
 
Theoremceqsex8v 2900* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)
   &       &       &       &       &       &       &       &       &       &       &       &       &       &       &       =>   
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