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

Statement | ||

Theorem | euxfr2 2901* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euxfr 2902* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euind 2903* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |

Theorem | reurex 2904 | Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.) |

Theorem | reu5 2905 | Restricted uniqueness in terms of "at most one." (Contributed by NM, 23-May-1999.) |

Theorem | reu2 2906* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |

Theorem | reu6 2907* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |

Theorem | reu3 2908* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu6i 2909* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | eqreu 2910* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | rmo4 2911* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu4 2912* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |

Theorem | reu7 2913* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu8 2914* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reueq 2915* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |

Theorem | 2reuswap 2916* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 14-Nov-2004.) |

Theorem | reuind 2917* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |

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 2922 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 2927 and cdeqab 2925. | ||

Syntax | wcdeq 2918 | 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 2919 | 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 2920 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqri 2921 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqth 2922 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqnot 2923 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal 2924* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab 2925* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal1 2926* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab1 2927* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqim 2928 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqcv 2929 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqeq 2930 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqel 2931 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | nfcdeq 2932* | 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 2933* | Variation of nfcdeq 2932 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

2.1.8 Russell's Paradox | ||

Theorem | ru 2934 |
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 4098 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 4090, Pairing prex 4155, Union uniex 4453, Power Set pwex 4131, and Infinity omex 7277 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 5233 (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 8035 and Cantor's Theorem canth 6225 are provably false! (See
ncanth 6226 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 4081
replaces ax-rep 4071) with ax-sep 4081 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 7244 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7247). See ruALT 7248 for an alternate proof of ru 2934 derived from that fact. (Contributed by NM, 7-Aug-1994.) |

2.1.9 Proper substitution of classes for
sets | ||

Syntax | wsbc 2935 | 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 2936 |
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 2961 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 2937 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 2943 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2937. The related definition df-csb 3024 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 2937 |
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 2936 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 2938 instead of df-sbc 2936. (dfsbcq2 2938 is needed because
unlike Quine we do not overload the df-sb 1884 syntax.) As a consequence of
these theorems, we can derive sbc8g 2942, which is a weaker version of
df-sbc 2936 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2942, so we will allow direct use of df-sbc 2936 after theorem sbc2or 2943 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 2938 | 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 1884 and substitution for class variables df-sbc 2936. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2937. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbsbc 2939 | Show that df-sb 1884 and df-sbc 2936 are equivalent when the class term in df-sbc 2936 is a set variable. This theorem lets us reuse theorems based on df-sb 1884 for proofs involving df-sbc 2936. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |

Theorem | sbceq1d 2940 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbceq1dd 2941 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbc8g 2942 | This is the closest we can get to df-sbc 2936 if we start from dfsbcq 2937 (see its comments) and dfsbcq2 2938. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |

Theorem | sbc2or 2943* | 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 2959 (false) and sbc6 2961 (true) conclusions. This is interesting since dfsbcq 2937 and dfsbcq2 2938 (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 2944 | 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 2945 | Equality theorem for class substitution. Class version of sbequ12 1893. (Contributed by NM, 26-Sep-2003.) |

Theorem | sbceq2a 2946 | Equality theorem for class substitution. Class version of sbequ12r 1894. (Contributed by NM, 4-Jan-2017.) |

Theorem | a4sbc 2947 | 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 1897 and ra4sbc 3013. (Contributed by NM, 16-Jan-2004.) |

Theorem | a4sbcd 2948 | 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 1897 and ra4sbc 3013. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbcth 2949 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |

Theorem | sbcthdv 2950* | Deduction version of sbcth 2949. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbcid 2951 | An identity theorem for substitution. See sbid 1896. (Contributed by Mario Carneiro, 18-Feb-2017.) |

Theorem | nfsbc1d 2952 | Deduction version of nfsbc1 2953. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1 2953 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1v 2954* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbcd 2955 | Deduction version of nfsbc 2956. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc 2956 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbcco 2957* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcco2 2958* | A composition law for class substitution. Importantly, may occur free in the class expression substituted for . (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbc5 2959* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbc6g 2960* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbc6 2961* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |

Theorem | sbc7 2962* | An equivalence for class substitution in the spirit of df-clab 2243. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | cbvsbc 2963 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | cbvsbcv 2964* | Change the bound variable of a class substitution using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbciegft 2965* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2966.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbciegf 2966* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcieg 2967* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |

Theorem | sbcie2g 2968* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 2969 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |

Theorem | sbcie 2969* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |

Theorem | sbciedf 2970* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcied 2971* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |

Theorem | sbcied2 2972* | Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) |

Theorem | elrabsf 2973 | Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 2873 has implicit substitution). The hypothesis specifies that must not be a free variable in . (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) |

Theorem | eqsbc3 2974* | Substitution applied to an atomic wff. Set theory version of eqsb3 2357. (Contributed by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcng 2975 | Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcimg 2976 | Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcan 2977 | Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcang 2978 | Distribution of class substitution over conjunction. (Contributed by NM, 21-May-2004.) |

Theorem | sbcor 2979 | Distribution of class substitution over disjunction. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcorg 2980 | Distribution of class substitution over disjunction. (Contributed by NM, 21-May-2004.) |

Theorem | sbcbig 2981 | Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.) |

Theorem | sbcal 2982* | Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbcalg 2983* | Move universal quantifier in and out of class substitution. (Contributed by NM, 16-Jan-2004.) |

Theorem | sbcex2 2984* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |

Theorem | sbcexg 2985* | Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) |

Theorem | sbceqal 2986* | Set theory version of sbeqal1 26929. (Contributed by Andrew Salmon, 28-Jun-2011.) |

Theorem | sbeqalb 2987* | Theorem *14.121 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 28-Jun-2011.) (Proof shortened by Wolf Lammen, 9-May-2013.) |

Theorem | sbcbid 2988 | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcbidv 2989* | Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.) |

Theorem | sbcbii 2990 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) |

Theorem | sbcbiiOLD 2991 | Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.) (New usage is discouraged.) |

Theorem | eqsbc3r 2992* | eqsbc3 2974 with set variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD 27629 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.) |

Theorem | sbc3ang 2993 | Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcel1gv 2994* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcel2gv 2995* | Class substitution into a membership relation. (Contributed by NM, 17-Nov-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbcimdv 2996* | Substitution analog of Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 11-Nov-2005.) |

Theorem | sbctt 2997 | Substitution for a variable not free in a wff does not affect it. (Contributed by Mario Carneiro, 14-Oct-2016.) |

Theorem | sbcgf 2998 | Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |

Theorem | sbc19.21g 2999 | Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.) |

Theorem | sbcg 3000* | Substitution for a variable not occurring in a wff does not affect it. Distinct variable form of sbcgf 2998. (Contributed by Alan Sare, 10-Nov-2012.) |

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