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Type | Label | Description |
---|---|---|

Statement | ||

Theorem | ceqsrexv 2901* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.) |

Theorem | ceqsrexbv 2902* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) |

Theorem | ceqsrex2v 2903* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.) |

Theorem | clel2 2904* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |

Theorem | clel3g 2905* | An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.) |

Theorem | clel3 2906* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |

Theorem | clel4 2907* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |

Theorem | pm13.183 2908* | 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 2909* | Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3547 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.) |

Theorem | rr19.28v 2910* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3549 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.) |

Theorem | elabgt 2911* | Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2915.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | elabgf 2912 | 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 2913* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | elab 2914* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) |

Theorem | elabg 2915* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) |

Theorem | elab2g 2916* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |

Theorem | elab2 2917* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |

Theorem | elab4g 2918* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |

Theorem | elab3gf 2919 | Membership in a class abstraction, with a weaker antecedent than elabgf 2912. (Contributed by NM, 6-Sep-2011.) |

Theorem | elab3g 2920* | Membership in a class abstraction, with a weaker antecedent than elabg 2915. (Contributed by NM, 29-Aug-2006.) |

Theorem | elab3 2921* | Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |

Theorem | elrabf 2922 | 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 | elrab 2923* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) |

Theorem | elrab3 2924* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |

Theorem | elrab2 2925* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |

Theorem | ralab 2926* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | ralrab 2927* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | rexab 2928* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab 2929* | Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralab2 2930* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralrab2 2931* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexab2 2932* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab2 2933* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | abidnf 2934* | 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 2935* | A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1756 and nfab 2423 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 2934 is useful. (Contributed by NM, 8-Dec-2006.) |

Theorem | eqeu 2936* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |

Theorem | eueq 2937* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |

Theorem | eueq1 2938* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq2 2939* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq3 2940* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |

Theorem | moeq 2941* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |

Theorem | moeq3 2942* | "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.) |

Theorem | mosub 2943* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |

Theorem | mo2icl 2944* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |

Theorem | mob2 2945* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |

Theorem | moi2 2946* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |

Theorem | mob 2947* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | moi 2948* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | morex 2949* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |

Theorem | euxfr2 2950* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euxfr 2951* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euind 2952* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |

Theorem | reu2 2953* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |

Theorem | reu6 2954* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |

Theorem | reu3 2955* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu6i 2956* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | eqreu 2957* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | rmo4 2958* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |

Theorem | reu4 2959* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |

Theorem | reu7 2960* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu8 2961* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reueq 2962* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |

Theorem | rmoan 2963 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |

Theorem | rmoim 2964 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | rmoimia 2965 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | rmoimi2 2966 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reuswap 2967* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |

Theorem | reuind 2968* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |

Theorem | 2rmorex 2969* | Double restricted quantification with "at most one," analogous to 2moex 2214. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem1 2970* | Lemma for 2reu5 2973. Note that does not mean "there is exactly one in and exactly one in such that holds;" see comment for 2eu5 2227. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem2 2971* | Lemma for 2reu5 2973. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem3 2972* | Lemma for 2reu5 2973. This lemma is interesting in its own right, showing that existential restriction in the last conjunct (the "at most one" part) is optional; compare rmo2 3076. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5 2973* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2227 and reu3 2955. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

2.1.7 Conditional equality
(experimental)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 2978 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 Otherwise, it is a primitive operation applied to smaller expressions. In these cases, for each set 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 2983 and cdeqab 2981. | ||

Syntax | wcdeq 2974 | 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 2975 | 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 2976 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqri 2977 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqth 2978 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqnot 2979 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal 2980* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab 2981* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal1 2982* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab1 2983* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqim 2984 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqcv 2985 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqeq 2986 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqel 2987 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | nfcdeq 2988* | 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 2989* | Variation of nfcdeq 2988 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

2.1.8 Russell's Paradox | ||

Theorem | ru 2990 |
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 ssex 4158 asserting that is a set only when it is smaller than some other set . However, Zermelo was then faced with a "chicken and egg" problem of how to show is a set, leading him to introduce the set-building axioms of Null Set 0ex 4150, Pairing prex 4217, Union uniex 4516, Power Set pwex 4193, and Infinity omex 7344 to give him some starting sets to work with (all of which, before Russell's Paradox, were immediate consequences of Frege's Comprehension). In 1922 Fraenkel strengthened the Subset Axiom with our present Replacement Axiom funimaex 5330 (whose modern formalization is due to Skolem, also in 1922). Thus in a very real sense Russell's Paradox spawned the invention of ZF set theory and completely revised the foundations of mathematics! Another mainstream formalization of set theory, devised by von Neumann, Bernays, and Goedel, uses class variables rather than set variables as its primitives. The axiom system NBG in [Mendelson] p. 225 is suitable for a Metamath encoding. NBG is a conservative extension of ZF in that it proves exactly the same theorems as ZF that are expressible in the language of ZF. An advantage of NBG is that it is finitely axiomatizable - the Axiom of Replacement can be broken down into a finite set of formulas that eliminate its wff metavariable. Finite axiomatizability is required by some proof languages (although not by Metamath). There is a stronger version of NBG called Morse-Kelley (axiom system MK in [Mendelson] p. 287).
Russell himself continued in a different direction, avoiding the paradox
with his "theory of types." Quine extended Russell's ideas to
formulate
his New Foundations set theory (axiom system NF of [Quine] p. 331). In
NF, the collection of all sets is a set, contradicting ZF and NBG set
theories, and it has other bizarre consequences: when sets become too
huge (beyond the size of those used in standard mathematics), the Axiom
of Choice ac4 8102 and Cantor's Theorem canth 6294 are provably false! (See
ncanth 6295 for some intuition behind the latter.)
Recent results (as of
2014) seem to show that NF is equiconsistent to Z (ZF in which ax-sep 4141
replaces ax-rep 4131) with ax-sep 4141 restricted to only bounded
quantifiers. NF is finitely axiomatizable and can be encoded in
Metamath using the axioms from T. Hailperin, "A set of axioms for
logic," Under our ZF set theory, every set is a member of the Russell class by elirrv 7311 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7314). See ruALT 7315 for an alternate proof of ru 2990 derived from that fact. (Contributed by NM, 7-Aug-1994.) |

2.1.9 Proper substitution of classes for
sets | ||

Syntax | wsbc 2991 | Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class for set variable in wff ." |

Definition | df-sbc 2992 |
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 3017 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 2993 below). For example, if is a proper class, Quine's substitution of for in evaluates to rather than our falsehood. (This can be seen by substituting , , and for for alpha, beta, and gamma in Subcase 1 of Quine's discussion on p. 42.) 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 The theorem sbc2or 2999 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2993. The related definition df-csb 3082 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 2993 |
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 2992 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 2994 instead of df-sbc 2992. (dfsbcq2 2994 is needed because
unlike Quine we do not overload the df-sb 1630 syntax.) As a consequence of
these theorems, we can derive sbc8g 2998, which is a weaker version of
df-sbc 2992 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2998, so we will allow direct use of df-sbc 2992 after theorem sbc2or 2999 below. 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 2994 | 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 1630 and substitution for class variables df-sbc 2992. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2993. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbsbc 2995 | Show that df-sb 1630 and df-sbc 2992 are equivalent when the class term in df-sbc 2992 is a set variable. This theorem lets us reuse theorems based on df-sb 1630 for proofs involving df-sbc 2992. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |

Theorem | sbceq1d 2996 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbceq1dd 2997 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbc8g 2998 | This is the closest we can get to df-sbc 2992 if we start from dfsbcq 2993 (see its comments) and dfsbcq2 2994. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |

Theorem | sbc2or 2999* | The disjunction of two equivalences for class substitution does not require a class existence hypothesis. This theorem tells us that there are only 2 possibilities for behavior at proper classes, matching the sbc5 3015 (false) and sbc6 3017 (true) conclusions. This is interesting since dfsbcq 2993 and dfsbcq2 2994 (from which it is derived) do not appear to say anything obvious about proper class behavior. Note that this theorem doesn't tell us that it is always one or the other at proper classes; it could "flip" between false (the first disjunct) and true (the second disjunct) as a function of some other variable that or may contain. (Contributed by NM, 11-Oct-2004.) (Proof modification is discouraged.) |

Theorem | sbcex 3000 | 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.) |

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