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Theorem List for Metamath Proof Explorer - 2801-2900   *Has distinct variable group(s)
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Statement

2.1.6  The universal class

Syntaxcvv 2801 Extend class notation to include the universal class symbol.

Theoremvjust 2802 Soundness justification theorem for df-v 2803. (Contributed by Rodolfo Medina, 27-Apr-2010.)

Definitiondf-v 2803 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 2804 All set variables are sets (see isset 2805). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.)

Theoremisset 2805* Two ways to say " is a set": A class is a member of the universal class (see df-v 2803) 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 4532. Note the when is not a set, it is called a proper class. In some theorems, such as uniexg 4533, 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 2292 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 2806 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.)

Theoremisseti 2807* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremissetri 2808* A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.)

Theoremelex 2809 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 2810 If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.)

Theoremelisset 2811* An element of a class exists. (Contributed by NM, 1-May-1995.)

Theoremelex22 2812* If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremelex2 2813* If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)

Theoremralv 2814 A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremrexv 2815 An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Theoremreuv 2816 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.)

Theoremrmov 2817 A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Theoremrabab 2818 A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremralcom4 2819* Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4 2820* Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrexcom4a 2821* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremrexcom4b 2822* Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)

Theoremceqsalt 2823* Closed theorem version of ceqsalg 2825. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsralt 2824* Restricted quantifier version of ceqsalt 2823. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsalg 2825* 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.)

Theoremceqsal 2826* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsalv 2827* A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.)

Theoremceqsralv 2828* Restricted quantifier version of ceqsalv 2827. (Contributed by NM, 21-Jun-2013.)

Theoremgencl 2829* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem2gencl 2830* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theorem3gencl 2831* Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.)

Theoremcgsexg 2832* Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)

Theoremcgsex2g 2833* Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)

Theoremcgsex4g 2834* An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.)

Theoremceqsex 2835* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremceqsexv 2836* Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.)

Theoremceqsex2 2837* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex2v 2838* Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.)

Theoremceqsex3v 2839* Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)

Theoremceqsex4v 2840* Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremceqsex6v 2841* Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)

Theoremceqsex8v 2842* Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Theoremgencbvex 2843* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremgencbvex2 2844* Restatement of gencbvex 2843 with weaker hypotheses. (Contributed by Jeffrey Hankins, 6-Dec-2006.)

Theoremgencbval 2845* Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.)

Theoremsbhypf 2846* Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3129. (Contributed by Raph Levien, 10-Apr-2004.)

Theoremvtoclgft 2847 Closed theorem form of vtoclgf 2855. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

Theoremvtocldf 2848 Implicit substitution of a class for a set variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtocld 2849* Implicit substitution of a class for a set variable. (Contributed by Mario Carneiro, 15-Oct-2016.)

Theoremvtoclf 2850* Implicit substitution of a class for a set variable. This is a generalization of chvar 1939. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl 2851* Implicit substitution of a class for a set variable. (Contributed by NM, 30-Aug-1993.)

Theoremvtocl2 2852* Implicit substitution of classes for set variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtocl3 2853* Implicit substitution of classes for set variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremvtoclb 2854* Implicit substitution of a class for a set variable. (Contributed by NM, 23-Dec-1993.)

Theoremvtoclgf 2855 Implicit substitution of a class for a set variable, with bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Proof shortened by Mario Carneiro, 10-Oct-2016.)

Theoremvtoclg 2856* Implicit substitution of a class expression for a set variable. (Contributed by NM, 17-Apr-1995.)

Theoremvtoclbg 2857* Implicit substitution of a class for a set variable. (Contributed by NM, 29-Apr-1994.)

Theoremvtocl2gf 2858 Implicit substitution of a class for a set variable. (Contributed by NM, 25-Apr-1995.)

Theoremvtocl3gf 2859 Implicit substitution of a class for a set variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtocl2g 2860* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 25-Apr-1995.)

Theoremvtoclgaf 2861* Implicit substitution of a class for a set variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)

Theoremvtoclga 2862* Implicit substitution of a class for a set variable. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl2gaf 2863* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 10-Aug-2013.)

Theoremvtocl2ga 2864* Implicit substitution of 2 classes for 2 set variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocl3gaf 2865* Implicit substitution of 3 classes for 3 set variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtocl3ga 2866* Implicit substitution of 3 classes for 3 set variables. (Contributed by NM, 20-Aug-1995.)

Theoremvtocleg 2867* Implicit substitution of a class for a set variable. (Contributed by NM, 10-Jan-2004.)

Theoremvtoclegft 2868* Implicit substitution of a class for a set variable. (Closed theorem version of vtoclef 2869.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremvtoclef 2869* Implicit substitution of a class for a set variable. (Contributed by NM, 18-Aug-1993.)

Theoremvtocle 2870* Implicit substitution of a class for a set variable. (Contributed by NM, 9-Sep-1993.)

Theoremvtoclri 2871* Implicit substitution of a class for a set variable. (Contributed by NM, 21-Nov-1994.)

Theoremspcimgft 2872 A closed version of spcimgf 2874. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgft 2873 A closed version of spcgf 2876. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)

Theoremspcimgf 2874 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcimegf 2875 Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgf 2876 Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 2-Feb-1997.) (Revised by Andrew Salmon, 12-Aug-2011.)

Theoremspcegf 2877 Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.)

Theoremspcimdv 2878* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcdv 2879* Rule of specialization, using implicit substitution. Analogous to rspcdv 2900. (Contributed by David Moews, 1-May-2017.)

Theoremspcimedv 2880* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremspcgv 2881* Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.)

Theoremspcegv 2882* Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)

Theoremspc2egv 2883* Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremspc2gv 2884* Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Theoremspc3egv 2885* Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspc3gv 2886* Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)

Theoremspcv 2887* Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Theoremspcev 2888* Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Theoremspc2ev 2889* Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Theoremrspct 2890* A closed version of rspc 2891. (Contributed by Andrew Salmon, 6-Jun-2011.)

Theoremrspc 2891* Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspce 2892* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Theoremrspcv 2893* Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspccv 2894* Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)

Theoremrspcva 2895* Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)

Theoremrspccva 2896* Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Theoremrspcev 2897* Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.)

Theoremrspcimdv 2898* Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcimedv 2899* Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Theoremrspcdv 2900* Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)

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