Home | Metamath
Proof ExplorerTheorem List
(p. 31 of 323)
| < 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
(1-21811) |
Hilbert Space Explorer
(21812-23334) |
Users' Mathboxes
(23335-32225) |

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

Statement | ||

Theorem | elab2g 3001* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |

Theorem | elab2 3002* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.) |

Theorem | elab4g 3003* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 17-Oct-2012.) |

Theorem | elab3gf 3004 | Membership in a class abstraction, with a weaker antecedent than elabgf 2997. (Contributed by NM, 6-Sep-2011.) |

Theorem | elab3g 3005* | Membership in a class abstraction, with a weaker antecedent than elabg 3000. (Contributed by NM, 29-Aug-2006.) |

Theorem | elab3 3006* | Membership in a class abstraction using implicit substitution. (Contributed by NM, 10-Nov-2000.) |

Theorem | elrabi 3007* | Implication for the membership in a restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017.) |

Theorem | elrabf 3008 | 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 3009* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.) |

Theorem | elrab3 3010* | Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.) |

Theorem | elrab2 3011* | Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |

Theorem | ralab 3012* | Universal quantification over a class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | ralrab 3013* | Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |

Theorem | rexab 3014* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 23-Jan-2014.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab 3015* | Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralab2 3016* | Universal quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | ralrab2 3017* | Universal quantification over a restricted class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexab2 3018* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | rexrab2 3019* | Existential quantification over a class abstraction. (Contributed by Mario Carneiro, 3-Sep-2015.) |

Theorem | abidnf 3020* | 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 3021* | A deduction theorem for converting the inference => into a closed theorem. Use nfa1 1794 and nfab 2506 to eliminate the hypothesis of the substitution instance of the inference. For converting the inference form into a deduction form, abidnf 3020 is useful. (Contributed by NM, 8-Dec-2006.) |

Theorem | eqeu 3022* | A condition which implies existential uniqueness. (Contributed by Jeff Hankins, 8-Sep-2009.) |

Theorem | eueq 3023* | Equality has existential uniqueness. (Contributed by NM, 25-Nov-1994.) |

Theorem | eueq1 3024* | Equality has existential uniqueness. (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq2 3025* | Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995.) |

Theorem | eueq3 3026* | Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.) |

Theorem | moeq 3027* | There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.) |

Theorem | moeq3 3028* | "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 3029* | "At most one" remains true after substitution. (Contributed by NM, 9-Mar-1995.) |

Theorem | mo2icl 3030* | Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.) |

Theorem | mob2 3031* | Consequence of "at most one." (Contributed by NM, 2-Jan-2015.) |

Theorem | moi2 3032* | Consequence of "at most one." (Contributed by NM, 29-Jun-2008.) |

Theorem | mob 3033* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | moi 3034* | Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.) |

Theorem | morex 3035* | Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.) |

Theorem | euxfr2 3036* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euxfr 3037* | Transfer existential uniqueness from a variable to another variable contained in expression . (Contributed by NM, 14-Nov-2004.) |

Theorem | euind 3038* | Existential uniqueness via an indirect equality. (Contributed by NM, 11-Oct-2010.) |

Theorem | reu2 3039* | A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.) |

Theorem | reu6 3040* | A way to express restricted uniqueness. (Contributed by NM, 20-Oct-2006.) |

Theorem | reu3 3041* | A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu6i 3042* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | eqreu 3043* | A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |

Theorem | rmo4 3044* | Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by NM, 16-Jun-2017.) |

Theorem | reu4 3045* | Restricted uniqueness using implicit substitution. (Contributed by NM, 23-Nov-1994.) |

Theorem | reu7 3046* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reu8 3047* | Restricted uniqueness using implicit substitution. (Contributed by NM, 24-Oct-2006.) |

Theorem | reueq 3048* | Equality has existential uniqueness. (Contributed by Mario Carneiro, 1-Sep-2015.) |

Theorem | rmoan 3049 | Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.) |

Theorem | rmoim 3050 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | rmoimia 3051 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | rmoimi2 3052 | Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reuswap 3053* | A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.) |

Theorem | reuind 3054* | Existential uniqueness via an indirect equality. (Contributed by NM, 16-Oct-2010.) |

Theorem | 2rmorex 3055* | Double restricted quantification with "at most one," analogous to 2moex 2288. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem1 3056* | Lemma for 2reu5 3059. Note that does not mean "there is exactly one in and exactly one in such that holds;" see comment for 2eu5 2301. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem2 3057* | Lemma for 2reu5 3059. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5lem3 3058* | Lemma for 2reu5 3059. 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 3162. (Contributed by Alexander van der Vekens, 17-Jun-2017.) |

Theorem | 2reu5 3059* | Double restricted existential uniqueness in terms of restricted existential quantification and restricted universal quantification, analogous to 2eu5 2301 and reu3 3041. (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 3064 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 3069 and cdeqab 3067. | ||

Syntax | wcdeq 3060 | 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 3061 | 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 3062 | Deduce conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqri 3063 | Property of conditional equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqth 3064 | Deduce conditional equality from a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqnot 3065 | Distribute conditional equality over negation. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal 3066* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab 3067* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqal1 3068* | Distribute conditional equality over quantification. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqab1 3069* | Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq | ||

Theorem | cdeqim 3070 | Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqcv 3071 | Conditional equality for set-to-class promotion. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

Theorem | cdeqeq 3072 | Distribute conditional equality over equality. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | cdeqel 3073 | Distribute conditional equality over elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq CondEq CondEq | ||

Theorem | nfcdeq 3074* | 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 3075* | Variation of nfcdeq 3074 for classes. (Contributed by Mario Carneiro, 11-Aug-2016.) |

CondEq | ||

2.1.8 Russell's Paradox | ||

Theorem | ru 3076 |
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 4260 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 4252, Pairing prex 4319, Union uniex 4619, Power Set pwex 4295, and Infinity omex 7491 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 5435 (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 8249 and Cantor's Theorem canth 6436 are provably false! (See
ncanth 6437 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 4243
replaces ax-rep 4233) with ax-sep 4243 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 7458 (derived from the Axiom of Regularity), so for us the Russell class equals the universe (theorem ruv 7461). See ruALT 7462 for an alternate proof of ru 3076 derived from that fact. (Contributed by NM, 7-Aug-1994.) |

2.1.9 Proper substitution of classes for
sets | ||

Syntax | wsbc 3077 | 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 3078 |
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 3103 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 3079 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 3085 shows the apparently "strongest" statement we can make regarding behavior at proper classes if we start from dfsbcq 3079. The related definition df-csb 3168 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 3079 |
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 3078 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 3080 instead of df-sbc 3078. (dfsbcq2 3080 is needed because
unlike Quine we do not overload the df-sb 1654 syntax.) As a consequence of
these theorems, we can derive sbc8g 3084, which is a weaker version of
df-sbc 3078 that leaves substitution undefined when is a proper class.
However, it is often a nuisance to have to prove the sethood hypothesis of sbc8g 3084, so we will allow direct use of df-sbc 3078 after theorem sbc2or 3085 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 3080 | 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 1654 and substitution for class variables df-sbc 3078. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3079. (Contributed by NM, 31-Dec-2016.) |

Theorem | sbsbc 3081 | Show that df-sb 1654 and df-sbc 3078 are equivalent when the class term in df-sbc 3078 is a set variable. This theorem lets us reuse theorems based on df-sb 1654 for proofs involving df-sbc 3078. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |

Theorem | sbceq1d 3082 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbceq1dd 3083 | Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbc8g 3084 | This is the closest we can get to df-sbc 3078 if we start from dfsbcq 3079 (see its comments) and dfsbcq2 3080. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.) |

Theorem | sbc2or 3085* | 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 3101 (false) and sbc6 3103 (true) conclusions. This is interesting since dfsbcq 3079 and dfsbcq2 3080 (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 3086 | 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 3087 | Equality theorem for class substitution. Class version of sbequ12 1931. (Contributed by NM, 26-Sep-2003.) |

Theorem | sbceq2a 3088 | Equality theorem for class substitution. Class version of sbequ12r 1932. (Contributed by NM, 4-Jan-2017.) |

Theorem | spsbc 3089 | 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 2037 and rspsbc 3155. (Contributed by NM, 16-Jan-2004.) |

Theorem | spsbcd 3090 | 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 2037 and rspsbc 3155. (Contributed by Mario Carneiro, 9-Feb-2017.) |

Theorem | sbcth 3091 | A substitution into a theorem remains true (when is a set). (Contributed by NM, 5-Nov-2005.) |

Theorem | sbcthdv 3092* | Deduction version of sbcth 3091. (Contributed by NM, 30-Nov-2005.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |

Theorem | sbcid 3093 | An identity theorem for substitution. See sbid 1934. (Contributed by Mario Carneiro, 18-Feb-2017.) |

Theorem | nfsbc1d 3094 | Deduction version of nfsbc1 3095. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1 3095 | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc1v 3096* | Bound-variable hypothesis builder for class substitution. (Contributed by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbcd 3097 | Deduction version of nfsbc 3098. (Contributed by NM, 23-Nov-2005.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | nfsbc 3098 | Bound-variable hypothesis builder for class substitution. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) |

Theorem | sbcco 3099* | A composition law for class substitution. (Contributed by NM, 26-Sep-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |

Theorem | sbcco2 3100* | 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.) |

< Previous Next > |

Copyright terms: Public domain | < Previous Next > |