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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cvv 2601 | Extend class notation to include the universal class symbol. |
Theorem | vjust 2602 | Soundness justification theorem for df-v 2603. (Contributed by Rodolfo Medina, 27-Apr-2010.) |
Definition | df-v 2603 | 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 2604 | All setvar variables are sets (see isset 2605). Theorem 6.8 of [Quine] p. 43. (Contributed by NM, 5-Aug-1993.) |
Theorem | isset 2605* |
Two ways to say "
is a set": A class is a member of the
universal class (see df-v 2603) 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 4192. Note the when is not a set,
it is called a proper class. In some theorems, such as uniexg 4193, 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 2077 requires that the expression substituted for not contain . (Also, the Metamath spec does not allow constants in the distinct variable list.) (Contributed by NM, 26-May-1993.) |
Theorem | issetf 2606 | 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 2607* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | issetri 2608* | A way to say " is a set" (inference rule). (Contributed by NM, 5-Aug-1993.) |
Theorem | eqvisset 2609 | A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2605 and issetri 2608. (Contributed by BJ, 27-Apr-2019.) |
Theorem | elex 2610 | 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 2611 | If a class is a member of another class, it is a set. (Contributed by NM, 11-Jun-1994.) |
Theorem | elexd 2612 | If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Theorem | elisset 2613* | An element of a class exists. (Contributed by NM, 1-May-1995.) |
Theorem | elex22 2614* | If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.) |
Theorem | elex2 2615* | If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) |
Theorem | ralv 2616 | A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | rexv 2617 | An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Theorem | reuv 2618 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by NM, 1-Nov-2010.) |
Theorem | rmov 2619 | A uniqueness quantifier restricted to the universe is unrestricted. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rabab 2620 | 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 2621* | Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4 2622* | Commutation of restricted and unrestricted existential quantifiers. (Contributed by NM, 12-Apr-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | rexcom4a 2623* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | rexcom4b 2624* | Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.) |
Theorem | ceqsalt 2625* | Closed theorem version of ceqsalg 2627. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsralt 2626* | Restricted quantifier version of ceqsalt 2625. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsalg 2627* | 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 2628* | 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 2629* | 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 2630* | Restricted quantifier version of ceqsalv 2629. (Contributed by NM, 21-Jun-2013.) |
Theorem | gencl 2631* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | 2gencl 2632* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | 3gencl 2633* | Implicit substitution for class with embedded variable. (Contributed by NM, 17-May-1996.) |
Theorem | cgsexg 2634* | Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.) |
Theorem | cgsex2g 2635* | Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.) |
Theorem | cgsex4g 2636* | An implicit substitution inference for 4 general classes. (Contributed by NM, 5-Aug-1995.) |
Theorem | ceqsex 2637* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | ceqsexv 2638* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) |
Theorem | ceqsex2 2639* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Theorem | ceqsex2v 2640* | Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) |
Theorem | ceqsex3v 2641* | Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.) |
Theorem | ceqsex4v 2642* | Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
Theorem | ceqsex6v 2643* | Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.) |
Theorem | ceqsex8v 2644* | Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.) |
Theorem | gencbvex 2645* | Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | gencbvex2 2646* | Restatement of gencbvex 2645 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.) |
Theorem | gencbval 2647* | Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.) |
Theorem | sbhypf 2648* | Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.) |
Theorem | vtoclgft 2649 | Closed theorem form of vtoclgf 2657. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | vtocldf 2650 | Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | vtocld 2651* | Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Theorem | vtoclf 2652* | Implicit substitution of a class for a setvar variable. This is a generalization of chvar 1680. (Contributed by NM, 30-Aug-1993.) |
Theorem | vtocl 2653* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993.) |
Theorem | vtocl2 2654* | Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | vtocl3 2655* | Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | vtoclb 2656* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 23-Dec-1993.) |
Theorem | vtoclgf 2657 | Implicit substitution of a class for a setvar 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.) |
Theorem | vtoclg 2658* | Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995.) |
Theorem | vtoclbg 2659* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.) |
Theorem | vtocl2gf 2660 | Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995.) |
Theorem | vtocl3gf 2661 | Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | vtocl2g 2662* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) |
Theorem | vtoclgaf 2663* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Theorem | vtoclga 2664* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995.) |
Theorem | vtocl2gaf 2665* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 10-Aug-2013.) |
Theorem | vtocl2ga 2666* | Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) |
Theorem | vtocl3gaf 2667* | Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Theorem | vtocl3ga 2668* | Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) |
Theorem | vtocleg 2669* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Jan-2004.) |
Theorem | vtoclegft 2670* | Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2671.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Theorem | vtoclef 2671* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 18-Aug-1993.) |
Theorem | vtocle 2672* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 9-Sep-1993.) |
Theorem | vtoclri 2673* | Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994.) |
Theorem | spcimgft 2674 | A closed version of spcimgf 2678. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | spcgft 2675 | A closed version of spcgf 2680. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.) |
Theorem | spcimegft 2676 | A closed version of spcimegf 2679. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | spcegft 2677 | A closed version of spcegf 2681. (Contributed by Jim Kingdon, 22-Jun-2018.) |
Theorem | spcimgf 2678 | Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | spcimegf 2679 | Existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | spcgf 2680 | 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.) |
Theorem | spcegf 2681 | Existential specialization, using implicit substitution. (Contributed by NM, 2-Feb-1997.) |
Theorem | spcimdv 2682* | Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | spcdv 2683* | Rule of specialization, using implicit substitution. Analogous to rspcdv 2704. (Contributed by David Moews, 1-May-2017.) |
Theorem | spcimedv 2684* | Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Theorem | spcgv 2685* | Rule of specialization, using implicit substitution. Compare Theorem 7.3 of [Quine] p. 44. (Contributed by NM, 22-Jun-1994.) |
Theorem | spcegv 2686* | Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.) |
Theorem | spc2egv 2687* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
Theorem | spc2gv 2688* | Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.) |
Theorem | spc3egv 2689* | Existential specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) |
Theorem | spc3gv 2690* | Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) |
Theorem | spcv 2691* | Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.) |
Theorem | spcev 2692* | Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.) |
Theorem | spc2ev 2693* | Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.) |
Theorem | rspct 2694* | A closed version of rspc 2695. (Contributed by Andrew Salmon, 6-Jun-2011.) |
Theorem | rspc 2695* | Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Theorem | rspce 2696* | Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.) |
Theorem | rspcv 2697* | Restricted specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) |
Theorem | rspccv 2698* | Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.) |
Theorem | rspcva 2699* | Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.) |
Theorem | rspccva 2700* | Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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