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Theorem List for Metamath Proof Explorer - 7401-7500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmarypha2lem4 7401* Lemma for marypha2 7402. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Theoremmarypha2 7402* Version of marypha1 7397 using a functional family of sets instead of a relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

2.4.38  Supremum

Syntaxcsup 7403 Extend class notation to include supremum of class . Here is ordinarily a relation that strictly orders class . For example, could be 'less than' and could be the set of real numbers.

Definitiondf-sup 7404* Define the supremum of class . It is meaningful when is a relation that strictly orders and when the supremum exists. For example, could be 'less than', could be the set of real numbers, and could be the set of all positive reals whose square is less than 2; in this case the supremum is defined as the square root of 2 per sqrval 11997. See dfsup2 7405 for alternate definition not requiring dummy variables.

We will also use this notation for "infimum" by replacing with . (Contributed by NM, 22-May-1999.)

Theoremdfsup2 7405 Quantifier free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)

Theoremdfsup2OLD 7406 Quantifier-free definition of supremum. (Contributed by Scott Fenton, 18-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremdfsup3OLD 7407 Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsupeq1 7408 Equality theorem for supremum. (Contributed by NM, 22-May-1999.)

Theoremsupeq1d 7409 Equality deduction for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremsupeq1i 7410 Equality inference for supremum. (Contributed by Paul Chapman, 22-Jun-2011.)

Theoremsupeq2 7411 Equality theorem for supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Theoremnfsup 7412 Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)

Theoremsupmo 7413* Any class has at most one supremum in (where is interpreted as 'less than'). (Contributed by NM, 5-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsupexd 7414 A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsupeu 7415* A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general). (Contributed by NM, 12-Oct-2004.)

Theoremsupval2 7416* Alternative expression for the supremum. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremeqsup 7417* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremeqsupd 7418* Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)

Theoremsupcl 7419* A supremum belongs to its base class (closure law). See also supub 7420 and suplub 7421. (Contributed by NM, 12-Oct-2004.)

Theoremsupub 7420* A supremum is an upper bound. See also supcl 7419 and suplub 7421.

This proof demonstrates how to expand an iota-based definition (df-iota 5377) using riotacl2 6522.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremsuplub 7421* A supremum is the least upper bound. See also supcl 7419 and supub 7420. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)

Theoremsuplub2 7422* Bidirectional form of suplub 7421. (Contributed by Mario Carneiro, 6-Sep-2014.)

Theoremsupnub 7423* An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004.)

Theoremsupex 7424 A supremum is a set. (Contributed by NM, 22-May-1999.)

Theoremsupmaxlem 7425* A set that contains the greatest element satisfies the antecedent in supremum theorems. This allows to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Theoremsupmax 7426* The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Theoremfisup2g 7427* A finite set satisfies the conditions to have a supremum. (Contributed by Mario Carneiro, 28-Apr-2015.)

Theoremfisupcl 7428 A nonempty finite set contains its supremum. (Contributed by Jeff Madsen, 9-May-2011.)

Theoremsuppr 7429 The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theoremsupsn 7430 The supremum of a singleton. (Contributed by NM, 2-Oct-2007.)

Theoremsupisolem 7431* Lemma for supiso 7433. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremsupisoex 7432* Lemma for supiso 7433. (Contributed by Mario Carneiro, 24-Dec-2016.)

Theoremsupiso 7433* Image of a supremum under an isomorphism. (Contributed by Mario Carneiro, 24-Dec-2016.)

2.4.39  Ordinal isomorphism, Hartog's theorem

Syntaxcoi 7434 Extend class definition to include the canonical order isomorphism to an ordinal.
OrdIso

Definitiondf-oi 7435* Define the canonical order isomorphism from the well-order on to an ordinal. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso Se recs recs

Theoremdfoi 7436* Rewrite df-oi 7435 with abbreviations. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs       OrdIso Se

Theoremoieq1 7437 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso OrdIso

Theoremoieq2 7438 Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso OrdIso

Theoremnfoi 7439 Hypothesis builder for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 15-Oct-2016.)
OrdIso

Theoremordiso2 7440 Generalize ordiso 7441 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.)

Theoremordiso 7441* Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.)

Theoremordtypecbv 7442* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 26-Jun-2015.)
recs                     recs

Theoremordtypelem1 7443* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem2 7444* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem3 7445* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem4 7446* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem5 7447* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 25-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem6 7448* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 24-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem7 7449* Lemma for ordtype 7457. is an initial segment of under the well-order . (Contributed by Mario Carneiro, 25-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem8 7450* Lemma for ordtype 7457. (Contributed by Mario Carneiro, 17-Oct-2009.)
recs                            OrdIso               Se

Theoremordtypelem9 7451* Lemma for ordtype 7457. Either the function OrdIso is an isomorphism onto all of , or OrdIso is not a set, which by oif 7455 implies that either is a proper class or . (Contributed by Mario Carneiro, 25-Jun-2015.)
recs                            OrdIso               Se

Theoremordtypelem10 7452* Lemma for ordtype 7457. Using ax-rep 4280, exclude the possibility that is a proper class and does not enumerate all of . (Contributed by Mario Carneiro, 25-Jun-2015.)
recs                            OrdIso               Se

Theoremoi0 7453 Definition of the ordinal isomorphism when its arguments are not meaningful. (Contributed by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremoicl 7454 The order type of the well-order on is an ordinal. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
OrdIso

Theoremoif 7455 The order isomorphism of the well-order on is a function. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso

Theoremoiiso2 7456 The order isomorphism of the well-order on is an isomorphism onto (which is a subset of by oif 7455). (Contributed by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremordtype 7457 For any set-like well-ordered class, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.) (Revised by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremoiiniseg 7458 is an initial segment of under the well-order . (Contributed by Mario Carneiro, 26-Jun-2015.)
OrdIso        Se

Theoremordtype2 7459 For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto isomorphically. Otherwise, is a proper class, which implies that either is a proper class or . This weak version of ordtype 7457 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremoiexg 7460 The order isomorphism on a set is a set. (Contributed by Mario Carneiro, 25-Jun-2015.)
OrdIso

Theoremoion 7461 The order type of the well-order on is an ordinal. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 23-May-2015.)
OrdIso

Theoremoiiso 7462 The order isomorphism of the well-order on is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso

Theoremoien 7463 The order type of a well-ordered set is equinumerous to the set. (Contributed by Mario Carneiro, 23-May-2015.)
OrdIso

Theoremoieu 7464 Uniqueness of the unique ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) (Revised by Mario Carneiro, 25-Jun-2015.)
OrdIso        Se

Theoremoismo 7465 When is a subclass of , is a strictly monotone ordinal functions, and it is also complete (it is an isomorphism onto all of ). The proof avoids ax-rep 4280 (the second statement is trivial under ax-rep 4280). (Contributed by Mario Carneiro, 26-Jun-2015.)
OrdIso

Theoremoiid 7466 The order type of an ordinal under the order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.)
OrdIso

Theoremhartogslem1 7467* Lemma for hartogs 7469. (Contributed by Mario Carneiro, 14-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
OrdIso

Theoremhartogslem2 7468* Lemma for hartogs 7469. (Contributed by Mario Carneiro, 14-Jan-2013.)
OrdIso

Theoremhartogs 7469* Given any set, the Hartogs number of the set is the least ordinal not dominated by that set. This theorem proves that there is always an ordinal which satisfies this. (This theorem can be proven trivially using the AC - see theorem ondomon 8394- but this proof works in ZF.) (Contributed by Jeff Hankins, 22-Oct-2009.) (Revised by Mario Carneiro, 15-May-2015.)

Theoremwofib 7470 The only sets which are well-ordered forwards and backwards are finite sets. (Contributed by Mario Carneiro, 30-Jan-2014.) (Revised by Mario Carneiro, 23-May-2015.)

Theoremwemaplem1 7471* Value of the lexicographic order on a sequence space. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwemaplem2 7472* Lemma for wemapso 7476. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremwemaplem3 7473* Lemma for wemapso 7476. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Theoremwemappo 7474* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Theoremwemapso2lem 7475* Lemma for wemapso 7476. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremwemapso 7476* Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Mario Carneiro, 8-Feb-2015.)

Theoremwemapso2 7477* An alternative to having a well-order on in wemapso 7476 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremcard2on 7478* Proof that the alternate definition cardval2 7834 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.)

Theoremcard2inf 7479* The definition cardval2 7834 has the curious property that for non-numerable sets (for which ndmfv 5714 yields ), it still evaluates to a non-empty set, and indeed it contains . (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

2.4.40  Hartogs function, order types, weak dominance

Syntaxchar 7480 Class symbol for the Hartogs/cardinal successor function.
har

Syntaxcwdom 7481 Class symbol for the weak dominance relation.
*

Definitiondf-har 7482* Define the Hartogs function , which maps all sets to the smallest ordinal that cannot be injected into the given set. In the important special case where is an ordinal, this is the cardinal successor operation.

Traditionally, the Hartogs number of a set is written and the cardinal successor ; we use functional notation for this, and cannot use the aleph symbol because it is taken for the enumerating function of the infinite initial ordinals df-aleph 7783.

Some authors define the Hartogs number of a set to be the least *infinite* ordinal which does not inject into it, thus causing the range to consist only of alephs. We use the simpler definition where the value can be any successor cardinal. (Contributed by Stefan O'Rear, 11-Feb-2015.)

har

Definitiondf-wdom 7483* A set is weakly dominated by a "larger" set iff the "larger" set can be mapped onto the "smaller" set or the smaller set is empty; equivalently if the smaller set can be placed into bijection with some partition of the larger set. When choice is assumed (as fodom 8358), this concides with the 1-1 defition df-dom 7070; however, it is not known whether this is a choice-equivalent or a strictly weaker form. Some discussion of this question can be found at http://boolesrings.org/asafk/2014/on-the-partition-principle/. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremharf 7484 Functionality of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har

Theoremharcl 7485 Closure of the Hartogs function in the ordinals. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har

Theoremharval 7486* Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
har

Theoremelharval 7487 The Hartogs number of a set is greater than all ordinals which inject into it. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
har

Theoremharndom 7488 The Hartogs number of a set does not inject into that set. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.)
har

Theoremharword 7489 Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
har har

Theoremrelwdom 7490 Weak dominance is a relation. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theorembrwdom 7491* Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theorembrwdomi 7492* Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
*

Theorembrwdomn0 7493* Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
*

Theorem0wdom 7494 Any set weakly dominates the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremfowdom 7495 An onto function implies weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremwdomref 7496 Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theorembrwdom2 7497* Alternate characterization of the weak dominance predicate which does not require special treatment of the empty set. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremdomwdom 7498 Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
*

Theoremwdomtr 7499 Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
* * *

Theoremwdomen1 7500 Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
* *

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