Home | Metamath
Proof ExplorerTheorem List
(p. 30 of 315)
| < Previous Next > |

Browser slow? Try the
Unicode version. |

Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs

Color key: | Metamath Proof Explorer
(1-21459) |
Hilbert Space Explorer
(21460-22982) |
Users' Mathboxes
(22983-31404) |

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

Statement | ||

Theorem | ralab 2901* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | ralrab 2902* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | rexab 2903* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab 2904* | Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralab2 2905* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralrab2 2906* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexab2 2907* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab2 2908* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | abidnf 2909* | 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 2910* | A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1719 and nfab 2398 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 2909 is useful. (Contributed by NM, 8-Dec-2006.) |

Theorem | eqeu 2911* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |

Theorem | eueq 2912* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |

Theorem | eueq1 2913* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq2 2914* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq3 2915* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |

Theorem | moeq 2916* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |

Theorem | moeq3 2917* | "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 2918* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |

Theorem | mo2icl 2919* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |

Theorem | mob2 2920* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |

Theorem | moi2 2921* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |

Theorem | mob 2922* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | moi 2923* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | morex 2924* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |

Theorem | euxfr2 2925* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euxfr 2926* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euind 2927* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |

Theorem | reu2 2928* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |

Theorem | reu6 2929* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |

Theorem | reu3 2930* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu6i 2931* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | eqreu 2932* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | rmo4 2933* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |

Theorem | reu4 2934* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |

Theorem | reu7 2935* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu8 2936* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reueq 2937* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |

Theorem | rmoan 2938 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |

Theorem | rmoim 2939 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | rmoimia 2940 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | rmoimi2 2941 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reuswap 2942* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |

Theorem | reuind 2943* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |

Theorem | 2rmorex 2944* | Double restricted quantification with "at most one," analogous to 2moex 2189. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem1 2945* | Lemma for 2reu5 2948. Note that does not mean "there is exactly one in and exactly one in such that holds;" see comment for 2eu5 2202. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem2 2946* | Lemma for 2reu5 2948. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem3 2947* | Lemma for 2reu5 2948. 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 3051. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5 2948* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2202 and reu3 2930. (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 2953 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 2958 and cdeqab 2956. | ||

Syntax | wcdeq 2949 | 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 2950 | 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 2951 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqri 2952 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqth 2953 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqnot 2954 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal 2955* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab 2956* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal1 2957* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab1 2958* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqim 2959 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqcv 2960 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqeq 2961 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqel 2962 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | nfcdeq 2963* | 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 2964* | Variation of nfcdeq 2963 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

2.1.8 Russell's Paradox | ||

Theorem | ru 2965 |
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 4132 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 4124, Pairing prex 4189, Union uniex 4488, Power Set pwex 4165, and Infinity omex 7312 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 5268 (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 8070 and Cantor's Theorem canth 6260 are provably false! (See
ncanth 6261 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 4115
replaces ax-rep 4105) with ax-sep 4115 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 7279 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7282). See ruALT 7283 for an alternate proof of ru 2965 derived from that fact. (Contributed by NM, 7-Aug-1994.) |

2.1.9 Proper substitution of classes for
sets | ||

Syntax | wsbc 2966 | 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 2967 |
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 2992 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 2968 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 2974 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 2968. The related definition df-csb 3057 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 2968 |
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 2967 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 2969 instead of df-sbc 2967. (dfsbcq2 2969 is needed because
unlike Quine we do not overload the df-sb 1884 syntax.) As a consequence of
these theorems, we can derive sbc8g 2973, which is a weaker version of
df-sbc 2967 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 2973, so we will allow direct use of df-sbc 2967 after theorem sbc2or 2974 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 2969 | 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 2967. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2968. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbsbc 2970 | Show that df-sb 1884 and df-sbc 2967 are equivalent when the class term in df-sbc 2967 is a set variable. This theorem lets us reuse theorems based on df-sb 1884 for proofs involving df-sbc 2967. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |

Theorem | sbceq1d 2971 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbceq1dd 2972 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbc8g 2973 | This is the closest we can get to df-sbc 2967 if we start from dfsbcq 2968 (see its comments) and dfsbcq2 2969. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |

Theorem | sbc2or 2974* | 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 2990 (false) and sbc6 2992 (true) conclusions. This is interesting since dfsbcq 2968 and dfsbcq2 2969 (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 2975 | 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 2976 | Equality theorem for class substitution. Class version of sbequ12 1893. (Contributed by NM, 26-Sep-2003.) |

Theorem | sbceq2a 2977 | Equality theorem for class substitution. Class version of sbequ12r 1894. (Contributed by NM, 4-Jan-2017.) |

Theorem | a4sbc 2978 | 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 3044. (Contributed by NM, 16-Jan-2004.) |

Theorem | a4sbcd 2979 | 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 3044. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbcth 2980 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |

Theorem | sbcthdv 2981* | Deduction version of sbcth 2980. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbcid 2982 | An identity theorem for substitution. See sbid 1896. (Contributed by Mario Carneiro, 18-Feb-2017.) |

Theorem | nfsbc1d 2983 | Deduction version of nfsbc1 2984. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1 2984 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1v 2985* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbcd 2986 | Deduction version of nfsbc 2987. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc 2987 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbcco 2988* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcco2 2989* | 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 2990* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbc6g 2991* | An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbc6 2992* | An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Proof shortened by Eric Schmidt, 17-Jan-2007.) |

Theorem | sbc7 2993* | An equivalence for class substitution in the spirit of df-clab 2245. Note that and don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | cbvsbc 2994 | Change bound variables in a wff substitution. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | cbvsbcv 2995* | 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 2996* | Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2997.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbciegf 2997* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcieg 2998* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.) |

Theorem | sbcie2g 2999* | Conversion of implicit substitution to explicit class substitution. This version of sbcie 3000 avoids a disjointness condition on by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |

Theorem | sbcie 3000* | Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 4-Sep-2004.) |

< Previous Next > |

Copyright terms: Public domain | < Previous Next > |