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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rmobiia 2801 | Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.) |
Theorem | rmobii 2802 | Formula-building rule for restricted existential quantifier (inference rule). (Contributed by NM, 16-Jun-2017.) |
Theorem | raleqf 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.) |
Theorem | rexeqf 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.) |
Theorem | reueq1f 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.) |
Theorem | rmoeq1f 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.) |
Theorem | raleq 2807* | Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
Theorem | rexeq 2808* | Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.) |
Theorem | reueq1 2809* | Equality theorem for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
Theorem | rmoeq1 2810* | Equality theorem for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | raleqi 2811* | Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | rexeqi 2812* | Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.) |
Theorem | raleqdv 2813* | Equality deduction for restricted universal quantifier. (Contributed by NM, 13-Nov-2005.) |
Theorem | rexeqdv 2814* | Equality deduction for restricted existential quantifier. (Contributed by NM, 14-Jan-2007.) |
Theorem | raleqbi1dv 2815* | Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
Theorem | rexeqbi1dv 2816* | Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.) |
Theorem | reueqd 2817* | Equality deduction for restricted uniqueness quantifier. (Contributed by NM, 5-Apr-2004.) |
Theorem | rmoeqd 2818* | Equality deduction for restricted uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | raleqbidv 2819* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Theorem | rexeqbidv 2820* | Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) |
Theorem | raleqbidva 2821* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
Theorem | rexeqbidva 2822* | Equality deduction for restricted universal quantifier. (Contributed by Mario Carneiro, 5-Jan-2017.) |
Theorem | mormo 2823 | Unrestricted "at most one" implies restricted "at most one". (Contributed by NM, 16-Jun-2017.) |
Theorem | reu5 2824 | Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) (Revised by NM, 16-Jun-2017.) |
Theorem | reurex 2825 | Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.) |
Theorem | reurmo 2826 | Restricted existential uniqueness implies restricted "at most one." (Contributed by NM, 16-Jun-2017.) |
Theorem | rmo5 2827 | Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) |
Theorem | nrexrmo 2828 | Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.) |
Theorem | cbvralf 2829 | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) |
Theorem | cbvrexf 2830 | Rule used to change bound variables, using implicit substitution. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 9-Oct-2016.) |
Theorem | cbvral 2831* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) |
Theorem | cbvrex 2832* | Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | cbvreu 2833* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | cbvrmo 2834* | Change the bound variable of restricted "at most one" using implicit substitution. (Contributed by NM, 16-Jun-2017.) |
Theorem | cbvralv 2835* | Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.) |
Theorem | cbvrexv 2836* | Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.) |
Theorem | cbvreuv 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.) |
Theorem | cbvrmov 2838* | Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | cbvraldva2 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.) |
Theorem | cbvrexdva2 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.) |
Theorem | cbvraldva 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.) |
Theorem | cbvrexdva 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.) |
Theorem | cbvral2v 2843* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by NM, 10-Aug-2004.) |
Theorem | cbvrex2v 2844* | Change bound variables of double restricted universal quantification, using implicit substitution. (Contributed by FL, 2-Jul-2012.) |
Theorem | cbvral3v 2845* | Change bound variables of triple restricted universal quantification, using implicit substitution. (Contributed by NM, 10-May-2005.) |
Theorem | cbvralsv 2846* | Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Theorem | cbvrexsv 2847* | Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Theorem | sbralie 2848* | Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.) |
Theorem | rabbiia 2849 | Equivalent wff's yield equal restricted class abstractions (inference rule). (Contributed by NM, 22-May-1999.) |
Theorem | rabbidva 2850* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 28-Nov-2003.) |
Theorem | rabbidv 2851* | Equivalent wff's yield equal restricted class abstractions (deduction rule). (Contributed by NM, 10-Feb-1995.) |
Theorem | rabeqf 2852 | Equality theorem for restricted class abstractions, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) |
Theorem | rabeq 2853* | Equality theorem for restricted class abstractions. (Contributed by NM, 15-Oct-2003.) |
Theorem | rabeqbidv 2854* | Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.) |
Theorem | rabeqbidva 2855* | Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Theorem | rabeq2i 2856 | Inference rule from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.) |
Theorem | cbvrab 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.) |
Theorem | cbvrabv 2858* | Rule to change the bound variable in a restricted class abstraction, using implicit substitution. (Contributed by NM, 26-May-1999.) |
Syntax | cvv 2859 | Extend class notation to include the universal class symbol. |
Theorem | vjust 2860 | Soundness justification theorem for df-v 2861. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
Definition | df-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.) |
Theorem | vex 2862 | All setvar variables are sets (see isset 2863). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.) |
Theorem | isset 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.) |
Theorem | issetf 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.) |
Theorem | isseti 2865* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | issetri 2866* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | elex 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.) |
Theorem | elexi 2868 | If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.) |
Theorem | elisset 2869* | An element of a class exists. (Contributed by NM, 1-May-1995.) |
Theorem | elex22 2870* | If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) |
Theorem | elex2 2871* | If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
Theorem | ralv 2872 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | rexv 2873 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | reuv 2874 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
Theorem | rmov 2875 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rabab 2876 | A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | ralcom4 2877* | Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4 2878* | Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4a 2879* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | rexcom4b 2880* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | ceqsalt 2881* | Closed theorem version of ceqsalg 2883. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsralt 2882* | Restricted quantifier version of ceqsalt 2881. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsalg 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.) |
Theorem | ceqsal 2884* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
Theorem | ceqsalv 2885* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993.) |
Theorem | ceqsralv 2886* | Restricted quantifier version of ceqsalv 2885. (Contributed by NM, 21-Jun-2013.) |
Theorem | gencl 2887* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | 2gencl 2888* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | 3gencl 2889* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | cgsexg 2890* | Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.) |
Theorem | cgsex2g 2891* | Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) |
Theorem | cgsex4g 2892* | An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) |
Theorem | ceqsex 2893* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsexv 2894* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) |
Theorem | ceqsex2 2895* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Theorem | ceqsex2v 2896* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Theorem | ceqsex3v 2897* | Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.) |
Theorem | ceqsex4v 2898* | Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
Theorem | ceqsex6v 2899* | Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.) |
Theorem | ceqsex8v 2900* | Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
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