Home | Intuitionistic Logic Explorer Theorem List (p. 28 of 106) | < Previous Next > |
Browser slow? Try the
Unicode version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | clel3g 2701* | An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.) |
Theorem | clel3 2702* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
Theorem | clel4 2703* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |
Theorem | pm13.183 2704* | Compare theorem *13.183 in [WhiteheadRussell] p. 178. Only is required to be a set. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Theorem | rr19.3v 2705* | Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.) |
Theorem | rr19.28v 2706* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.) |
Theorem | elabgt 2707* | Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2711.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
Theorem | elabgf 2708 | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | elabf 2709* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |
Theorem | elab 2710* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) |
Theorem | elabg 2711* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) |
Theorem | elab2g 2712* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
Theorem | elab2 2713* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |
Theorem | elab4g 2714* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |
Theorem | elab3gf 2715 | Membership in a class abstraction, with a weaker antecedent than elabgf 2708. (Contributed by NM, 6-Sep-2011.) |
Theorem | elab3g 2716* | Membership in a class abstraction, with a weaker antecedent than elabg 2711. (Contributed by NM, 29-Aug-2006.) |
Theorem | elab3 2717* | Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |
Theorem | elrabi 2718* | Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |
Theorem | elrabf 2719 | Membership in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) |
Theorem | elrab3t 2720* | Membership in a restricted class abstraction, using implicit substitution. (Closed theorem version of elrab3 2722.) (Contributed by Thierry Arnoux, 31-Aug-2017.) |
Theorem | elrab 2721* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) |
Theorem | elrab3 2722* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |
Theorem | elrab2 2723* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |
Theorem | ralab 2724* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Theorem | ralrab 2725* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Theorem | rexab 2726* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Theorem | rexrab 2727* | Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |
Theorem | ralab2 2728* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Theorem | ralrab2 2729* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Theorem | rexab2 2730* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Theorem | rexrab2 2731* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Theorem | abidnf 2732* | Identity used to create closed-form versions of bound-variable hypothesis builders for class expressions. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Mario Carneiro, 12-Oct-2016.) |
Theorem | dedhb 2733* | A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1450 and nfab 2198 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 2732 is useful. (Contributed by NM, 8-Dec-2006.) |
Theorem | eqeu 2734* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |
Theorem | eueq 2735* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |
Theorem | eueq1 2736* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |
Theorem | eueq2dc 2737* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |
DECID | ||
Theorem | eueq3dc 2738* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |
DECID DECID | ||
Theorem | moeq 2739* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |
Theorem | moeq3dc 2740* | "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.) |
DECID DECID | ||
Theorem | mosubt 2741* | "At most one" remains true after substitution. (Contributed by Jim Kingdon, 18-Jan-2019.) |
Theorem | mosub 2742* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |
Theorem | mo2icl 2743* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |
Theorem | mob2 2744* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |
Theorem | moi2 2745* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |
Theorem | mob 2746* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | moi 2747* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |
Theorem | morex 2748* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | euxfr2dc 2749* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
DECID | ||
Theorem | euxfrdc 2750* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |
DECID | ||
Theorem | euind 2751* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |
Theorem | reu2 2752* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |
Theorem | reu6 2753* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |
Theorem | reu3 2754* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu6i 2755* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | eqreu 2756* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Theorem | rmo4 2757* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |
Theorem | reu4 2758* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |
Theorem | reu7 2759* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reu8 2760* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |
Theorem | reueq 2761* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |
Theorem | rmoan 2762 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |
Theorem | rmoim 2763 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimia 2764 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | rmoimi2 2765 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | 2reuswapdc 2766* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |
DECID | ||
Theorem | reuind 2767* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |
Theorem | 2rmorex 2768* | Double restricted quantification with "at most one," analogous to 2moex 2002. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |
Theorem | nelrdva 2769* | Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.) |
This is a very useless definition, which "abbreviates" as CondEq . What this display hides, though, is that the first expression, even though it has a shorter constant string, is actually much more complicated in its parse tree: it is parsed as (wi (wceq (cv vx) (cv vy)) wph), while the CondEq version is parsed as (wcdeq vx vy wph). It also allows us to give a name to the specific 3-ary operation . This is all used as part of a metatheorem: we want to say that and are provable, for any expressions or in the language. The proof is by induction, so the base case is each of the primitives, which is why you will see a theorem for each of the set.mm primitive operations. The metatheorem comes with a disjoint variables assumption: every variable in is assumed disjoint from except itself. For such a proof by induction, we must consider each of the possible forms of . If it is a variable other than , then we have CondEq or CondEq , which is provable by cdeqth 2774 and reflexivity. Since we are only working with class and wff expressions, it can't be itself in set.mm, but if it was we'd have to also prove CondEq (where set equality is being used on the right). Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each setvar variable parameter to the operation, we must consider if it is equal to or not, which yields 2^n proof obligations. Luckily, all primitive operations in set.mm have either zero or one set variable, so we only need to prove one statement for the non-set constructors (like implication) and two for the constructors taking a set (the forall and the class builder). In each of the primitive proofs, we are allowed to assume that is disjoint from and vice versa, because this is maintained through the induction. This is how we satisfy the DV assumptions of cdeqab1 2779 and cdeqab 2777. | ||
Syntax | wcdeq 2770 | Extend wff notation to include conditional equality. This is a technical device used in the proof that is the not-free predicate, and that definitions are conservative as a result. |
CondEq | ||
Definition | df-cdeq 2771 | Define conditional equality. All the notation to the left of the is fake; the parentheses and arrows are all part of the notation, which could equally well be written CondEq. On the right side is the actual implication arrow. The reason for this definition is to "flatten" the structure on the right side (whose tree structure is something like (wi (wceq (cv vx) (cv vy)) wph) ) into just (wcdeq vx vy wph). (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqi 2772 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqri 2773 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqth 2774 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqnot 2775 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal 2776* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab 2777* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqal1 2778* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqab1 2779* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq | ||
Theorem | cdeqim 2780 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqcv 2781 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | cdeqeq 2782 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | cdeqel 2783 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq CondEq CondEq | ||
Theorem | nfcdeq 2784* | 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 2785* | Variation of nfcdeq 2784 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |
CondEq | ||
Theorem | ru 2786 |
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 3903. (Contributed by NM, 7-Aug-1994.) |
Syntax | wsbc 2787 | 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 2788 |
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 2812 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 2789 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 2789, which holds for both our definition and Quine's, and from which we can derive a weaker version of df-sbc 2788 in the form of sbc8g 2794. 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 2788 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 2789 |
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 2788 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 2790 instead of df-sbc 2788. (dfsbcq2 2790 is needed because
unlike Quine we do not overload the df-sb 1662 syntax.) As a consequence of
these theorems, we can derive sbc8g 2794, which is a weaker version of
df-sbc 2788 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2794, so we will allow direct use of df-sbc 2788. 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 2790 | 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 1662 and substitution for class variables df-sbc 2788. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2789. (Contributed by NM, 31-Dec-2016.) |
Theorem | sbsbc 2791 | Show that df-sb 1662 and df-sbc 2788 are equivalent when the class term in df-sbc 2788 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1662 for proofs involving df-sbc 2788. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Theorem | sbceq1d 2792 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbceq1dd 2793 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Theorem | sbc8g 2794 | This is the closest we can get to df-sbc 2788 if we start from dfsbcq 2789 (see its comments) and dfsbcq2 2790. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |
Theorem | sbcex 2795 | 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 2796 | Equality theorem for class substitution. Class version of sbequ12 1670. (Contributed by NM, 26-Sep-2003.) |
Theorem | sbceq2a 2797 | Equality theorem for class substitution. Class version of sbequ12r 1671. (Contributed by NM, 4-Jan-2017.) |
Theorem | spsbc 2798 | 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 1674 and rspsbc 2868. (Contributed by NM, 16-Jan-2004.) |
Theorem | spsbcd 2799 | 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 1674 and rspsbc 2868. (Contributed by Mario Carneiro, 9-Feb-2017.) |
Theorem | sbcth 2800 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |