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

Statement | ||

Theorem | ceqsexg 2901* | A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.) |

Theorem | ceqsexgv 2902* | Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) |

Theorem | ceqsrexv 2903* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.) |

Theorem | ceqsrexbv 2904* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) |

Theorem | ceqsrex2v 2905* | Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.) |

Theorem | clel2 2906* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |

Theorem | clel3g 2907* | An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.) |

Theorem | clel3 2908* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |

Theorem | clel4 2909* | An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.) |

Theorem | pm13.183 2910* | 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 2911* | Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3549 when there is an outer quantifier. (Contributed by NM, 25-Oct-2012.) |

Theorem | rr19.28v 2912* | Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3551 when there is an outer quantifier. (Contributed by NM, 29-Oct-2012.) |

Theorem | elabgt 2913* | Membership in a class abstraction, using implicit substitution. (Closed theorem version of elabg 2917.) (Contributed by NM, 7-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | elabgf 2914 | 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 2915* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | elab 2916* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.) |

Theorem | elabg 2917* | Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.) |

Theorem | elab2g 2918* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |

Theorem | elab2 2919* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |

Theorem | elab4g 2920* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |

Theorem | elab3gf 2921 | Membership in a class abstraction, with a weaker antecedent than elabgf 2914. (Contributed by NM, 6-Sep-2011.) |

Theorem | elab3g 2922* | Membership in a class abstraction, with a weaker antecedent than elabg 2917. (Contributed by NM, 29-Aug-2006.) |

Theorem | elab3 2923* | Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |

Theorem | elrabf 2924 | 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 2925* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) |

Theorem | elrab3 2926* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |

Theorem | elrab2 2927* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |

Theorem | ralab 2928* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | ralrab 2929* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | rexab 2930* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab 2931* | Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralab2 2932* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralrab2 2933* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexab2 2934* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab2 2935* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | abidnf 2936* | 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 2937* | A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1758 and nfab 2425 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 2936 is useful. (Contributed by NM, 8-Dec-2006.) |

Theorem | eqeu 2938* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |

Theorem | eueq 2939* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |

Theorem | eueq1 2940* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq2 2941* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq3 2942* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |

Theorem | moeq 2943* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |

Theorem | moeq3 2944* | "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 2945* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |

Theorem | mo2icl 2946* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |

Theorem | mob2 2947* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |

Theorem | moi2 2948* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |

Theorem | mob 2949* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | moi 2950* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | morex 2951* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |

Theorem | euxfr2 2952* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euxfr 2953* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euind 2954* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |

Theorem | reu2 2955* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |

Theorem | reu6 2956* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |

Theorem | reu3 2957* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu6i 2958* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | eqreu 2959* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | rmo4 2960* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |

Theorem | reu4 2961* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |

Theorem | reu7 2962* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu8 2963* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reueq 2964* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |

Theorem | rmoan 2965 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |

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

Theorem | rmoimia 2967 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | rmoimi2 2968 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reuswap 2969* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |

Theorem | reuind 2970* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |

Theorem | 2rmorex 2971* | Double restricted quantification with "at most one," analogous to 2moex 2216. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem1 2972* | Lemma for 2reu5 2975. Note that does not mean "there is exactly one in and exactly one in such that holds;" see comment for 2eu5 2229. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem2 2973* | Lemma for 2reu5 2975. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem3 2974* | Lemma for 2reu5 2975. 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 3078. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5 2975* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2229 and reu3 2957. (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 2980 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 2985 and cdeqab 2983. | ||

Syntax | wcdeq 2976 | 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 2977 | 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 2978 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqri 2979 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqth 2980 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqnot 2981 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

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

CondEq CondEq | ||

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

CondEq CondEq | ||

Theorem | cdeqal1 2984* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab1 2985* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

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

CondEq CondEq CondEq | ||

Theorem | cdeqcv 2987 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqeq 2988 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqel 2989 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | nfcdeq 2990* | 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 2991* | Variation of nfcdeq 2990 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

2.1.8 Russell's Paradox | ||

Theorem | ru 2992 |
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 4160 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 4152, Pairing prex 4219, Union uniex 4518, Power Set pwex 4195, and Infinity omex 7346 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 5332 (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 8104 and Cantor's Theorem canth 6296 are provably false! (See
ncanth 6297 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 4143
replaces ax-rep 4133) with ax-sep 4143 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 7313 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7316). See ruALT 7317 for an alternate proof of ru 2992 derived from that fact. (Contributed by NM, 7-Aug-1994.) |

2.1.9 Proper substitution of classes for
sets | ||

Syntax | wsbc 2993 | 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 2994 |
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 3019 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 2995 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 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 3001 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2995. The related definition df-csb 3084 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 2995 |
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 2994 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 2996 instead of df-sbc 2994. (dfsbcq2 2996 is needed because
unlike Quine we do not overload the df-sb 1632 syntax.) As a consequence of
these theorems, we can derive sbc8g 3000, which is a weaker version of
df-sbc 2994 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3000, so we will allow direct use of df-sbc 2994 after theorem sbc2or 3001 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 2996 | 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 1632 and substitution for class variables df-sbc 2994. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2995. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbsbc 2997 | Show that df-sb 1632 and df-sbc 2994 are equivalent when the class term in df-sbc 2994 is a set variable. This theorem lets us reuse theorems based on df-sb 1632 for proofs involving df-sbc 2994. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |

Theorem | sbceq1d 2998 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbceq1dd 2999 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbc8g 3000 | This is the closest we can get to df-sbc 2994 if we start from dfsbcq 2995 (see its comments) and dfsbcq2 2996. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |

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