mmtheorems57 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5601-5700 - Page 57 of 175

Type	Label	Description
Statement
Theorem	limensuc 5601	A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
Pigeonhole Principle
Theorem	phplem1 5602	Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element.
Theorem	phplem2 5603	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
Theorem	phplem3 5604	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
Theorem	phplem4 5605	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
Theorem	nneneq 5606	Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136.
|- ((A e. om /\ B e. om) -> (A ~~ B <-> A = B))
Theorem	php 5607	Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 5602 through phplem4 5605, nneneq 5606, and this final piece of the proof.
|- ((A e. om /\ B C. A) -> -. A ~~ B)
Theorem	php2 5608	Corollary of Pigeonhole Principle.
|- ((A e. om /\ B C. A) -> B ~< A)
Theorem	php3 5609	Corollary of Pigeonhole Principle. If A is finite and B is a proper subset of A, the B is strictly less numerous than A. Stronger version of Corollary 6C of [Enderton] p. 135.
|- ((A e. Fin /\ B C. A) -> B ~< A)
Theorem	php4 5610	Corollary of the Pigeonhole Principle php 5607: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
Theorem	php5 5611	Corollary of the Pigeonhole Principle php 5607: a natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90.
|- (A e. om -> -. A ~~ suc A)
Finite sets
Theorem	onomeneq 5612	An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse.
|- ((A e. On /\ B e. om) -> (A ~~ B <-> A = B))
Theorem	onfin 5613	An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92.
|- (A e. On -> (A e. Fin <-> A e. om))
Theorem	nndomo 5614	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146.
|- ((A e. om /\ B e. om) -> (A ~<_ B <-> A C_ B))
Theorem	nnsdomo 5615	Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A C. B))
Theorem	omsucdom 5616	Strict dominance of natural numbers is the same as dominance over the successor of the smaller.
|- ((A e. om /\ B e. om) -> (A ~< B <-> suc A ~<_ B))
Theorem	sucdomi 5617	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 5994.
|- ((A e. om /\ B e. C) -> (suc A ~<_ B -> A ~< B))
Theorem	0sdom1dom 5618	Strict dominance over zero is the same as dominance over one.
|- A e. _V   =>   |- ((/) ~< A <-> 1o ~<_ A)
Theorem	1sdom2 5619	Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
Theorem	finsucdom 5620	Strict dominance of a finite set over a natural number is the same as dominance over its successor.
|- ((A e. om /\ B e. Fin) -> (A ~< B <-> suc A ~<_ B))
Theorem	pssinf 5621	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136.
|- ((A C. B /\ A ~~ B) -> -. B e. Fin)
Theorem	ominf 5622	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. om e. Fin
Theorem	omsdomnn 5623	Omega strictly dominates a natural number. Example 3 of [Enderton] p. 146. Here we use A ~<_ om /\ -. om ~~ A instead of A ~< om because, due to a peculiarity ultimately caused our ordered pair definition, we would need the Axiom of infinity (which we have avoided up to now) in order to prove the latter.
|- (A e. om -> (A ~<_ om /\ -. om ~~ A))
Theorem	isfinite1 5624	Omega strictly dominates a finite set. See comment in omsdomnn 5623.
|- (A e. Fin -> (A ~<_ om /\ -. om ~~ A))
Theorem	infsdomnn 5625	An infinite set strictly dominates a natural number.
|- A e. _V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
Theorem	infn0 5626	An infinite set is not empty.
|- A e. _V   =>   |- (om ~<_ A -> A =/= (/))
Theorem	enfi 5627	Equinmerous sets have the same finiteness.
|- ((B e. C /\ A ~~ B) -> (A e. Fin <-> B e. Fin))
Theorem	pssnn 5628	A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137.
|- ((A e. om /\ B C. A) -> E.x e. A B ~~ x)
Theorem	ssnnfi 5629	A subset of a natural number is finite.
|- ((A e. om /\ B C_ A) -> B e. Fin)
Theorem	ssfi 5630	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((A e. Fin /\ B C_ A) -> B e. Fin)
Theorem	domfi 5631	A set dominated by a finite set is finite.
|- ((A e. Fin /\ B ~<_ A) -> B e. Fin)
Theorem	xpfi 5632	The components of a non-empty finite cross product are finite. (Contributed by Paul Chapman, 11-Apr-2009.)
|- (((A X. B) e. Fin /\ (A X. B) =/= (/)) -> (A e. Fin /\ B e. Fin))
Theorem	unblem1 5633	Lemma for unbnn 5637. After removing the successor of an element from an unbounded set of natural numbers, the intersection of the result belongs to the original unbounded set.
Theorem	unblem2 5634	Lemma for unbnn 5637. The value of the function F belongs to the unbounded set of natural numbers A.
Theorem	unblem3 5635	Lemma for unbnn 5637. The value of the function F is less than its value at a successor.
Theorem	unblem4 5636	Lemma for unbnn 5637. The function F maps the set of natural numbers one-to-one to the set of unbounded natural numbers A.
Theorem	unbnn 5637	Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. Part of the proof of Theorem 42 of [Suppes] p. 151. See unbnn3 5746 for a stronger version without the hypothesis.
|- A e. _V   =>   |- ((A C_ om /\ A.x e. om E.y e. A x e. y) -> A ~~ om)
Theorem	unbnn2 5638	Version of unbnn 5637 that does not require a strict upper bound.
|- A e. _V   =>   |- ((A C_ om /\ A.x e. om E.y e. A x C_ y) -> A ~~ om)
Theorem	isfinite2 5639	Any set strictly dominated by the class of natural numbers is finite. Sufficiency part of Theorem 42 of [Suppes] p. 151. This theorem does not require the Axiom of Infinity.
|- (A ~< om -> A e. Fin)
Theorem	fin2inf 5640	This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of strict dominance, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless om exists.
|- (A ~< om -> om e. _V)
Theorem	unfilem1 5641	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem2 5642	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem3 5643	Lemma for proving that the union of two finite sets is finite.
Theorem	unfi 5644	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144.
|- ((A e. Fin /\ B e. Fin) -> (A u. B) e. Fin)
Theorem	unfi2 5645	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 5644 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 5640).
|- ((A ~< om /\ B ~< om) -> (A u. B) ~< om)
Theorem	infcntss 5646	Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.)
|- A e. _V   =>   |- (om ~<_ A -> E.x(x C_ A /\ x ~~ om))
Theorem	prfi 5647	An unordered pair is finite.
|- {A, B} e. Fin
Theorem	unifi 5648	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144.
|- ((A e. Fin /\ A.x e. A x e. Fin) -> U.A e. Fin)
Theorem	unifi2 5649	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 5648 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 5640).
|- ((A ~< om /\ A.x e. A x ~< om) -> U.A ~< om)
Theorem	fiint 5650	Equivalent ways of stating the finite intersection property. We show two ways of saying, "the intersection of elements in every finite non-empty subcollection of A is in A." This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections. Some texts use the left-hand version of this axiom and others the right-hand version, but as our proof here shows, their "intuitively obvious" equivalence can be non-trivial to establish formally.
|- (A.x e. A A.y e. A (x i^i y) e. A <-> A.x((x C_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))
Theorem	abfii1 5651	Two ways to express the collection of finite intersections of a set A.
|- |^|{x | (A C_ x /\ A.y e. x A.z e. x (y i^i z) e. x)} = |^|{x | (A C_ x /\ A.y((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))}
Theorem	abfii2 5652	Two ways to express the collection of finite intersections of a set A.
|- A e. _V   =>   |- {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} = |^|{x | A.y((y C_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)}
Theorem	abfii3 5653	Two ways to express the collection of finite intersections of a set A.
|- A e. _V   =>   |- |^|{x | (A C_ x /\ A.y((y C_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))} = |^|{x | A.y((y C_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)}
Theorem	abfii4 5654	Two ways to express the collection of finite intersections of a set A. Even though the expressions differ by only one symbol, the proof is not simple.
|- A e. _V   =>   |- |^|{x | (A C_ x /\ A.y((y C_ x /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))} = |^|{x | (A C_ x /\ A.y((y C_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))}
Theorem	abfii5 5655	Two ways to express the collection of finite intersections of a set A.
|- A e. _V   =>   |- |^|{x | (A C_ x /\ A.y e. x A.z e. x (y i^i z) e. x)} = {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}
Theorem	fodomfi 5656	An onto function implies dominance of domain over range, for finite sets. Unlike fodom 5960 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof.
|- ((A e. Fin /\ F:A-onto->B) -> B ~<_ A)
Theorem	fodomfib 5657	Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 5962 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof.
|- (A e. Fin -> ((A =/= (/) /\ E.f f:A-onto->B) <-> ((/) ~< B /\ B ~<_ A)))
Theorem	fofi 5658	If a function has a finite domain, its range is finite. Theorem 37 of [Suppes] p. 104.
|- ((A e. Fin /\ F:A-onto->B) -> B e. Fin)
Theorem	iunfi 5659	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This is the indexed union version of unifi 5648. Note that B depends on x, i.e. can be thought of as B(x).
|- ((A e. Fin /\ A.x e. A B e. Fin) -> U_x e. A B e. Fin)
Theorem	pwfilem 5660	Lemma for pwfi 5661.
Theorem	pwfi 5661	The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105.
|- (A e. Fin <-> ~PA e. Fin)
Theorem	pm54.43 5662	Theorem *54.43 of [WhiteheadRussell] p. 360. "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." See http://en.wikipedia.org/wiki/Principia_Mathematica#Quotations. This theorem states that two sets of cardinality 1 are disjoint iff their union has cardinality 2. Whitehead and Russell define 1 as the collection of all sets with cardinality 1 (i.e. all singletons; see card1 5983), so that their A e. 1 means, in our notation, A e. {x | (card` x) = 1o} i.e. (card` A) = 1o (by elab 2403) i.e. A ~~ 1o (by carden 5981 and cardnn 5870). We do not have several of their earlier lemmas available (which would otherwise be unused by our different approach to arithmetic), so our proof is longer. (It is also longer because we must show every detail.) Theorem pm110.643 6072 shows the derivation of 1+1=2 for cardinal numbers from this theorem.
|- ((A ~~ 1o /\ B ~~ 1o) -> ((A i^i B) = (/) <-> (A u. B) ~~ 2o))
Supremum
Syntax	csup 5663	Extend class notation to include supremum of class A. Here R is ordinarily a relation that strictly orders class B. For example, R could be 'less than' and B could be the set of real numbers.
class sup(A, B, R)
Definition	df-sup 5664	Define the supremum of class B. It is meaningful when R is a relation that strictly orders A and when the supremum exists. For example, R could be 'less than', A could be the set of real numbers, and B 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 7921. We will also use this notation for "infimum" by replacing R with `'R.
|- sup(B, A, R) = U.{x e. A | (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))}
Theorem	supeq1 5665	Equality theorem for supremum.
|- (B = C -> sup(B, A, R) = sup(C, A, R))
Theorem	supmo 5666	Any class B has at most one supremum in A (where R is interpreted as 'less than').
|- R Or A   =>   |- E*x(x e. A /\ (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
Theorem	supex 5667	A supremum is a set.
|- R Or A   =>   |- sup(B, A, R) e. _V
Theorem	supeu 5668	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general).
|- R Or A   =>   |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
Theorem	supcl 5669	A supremum belongs to its base class (closure law).
|- R Or A   =>   |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> sup(B, A, R) e. A)
Theorem	supub 5670	A supremum is an upper bound.
|- R Or A   =>   |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> (C e. B -> -. sup(B, A, R)RC))
Theorem	suplub 5671	A supremum is the least upper bound.
|- R Or A   =>   |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((C e. A /\ CRsup(B, A, R)) -> E.z e. B CRz))
Theorem	supnub 5672	An upper bound is not less than the supremum.
|- R Or A   =>   |- (E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)) -> ((C e. A /\ A.z e. B -. CRz) -> -. CRsup(B, A, R)))
Theorem	supeui 5673	A supremum is unique. Similar to Theorem I.26 of [Apostol] p. 24 (but for suprema in general).
|- R Or A   &   |- E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))   =>   |- E!x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))
Theorem	supcli 5674	A supremum belongs to its base class (closure law).
|- R Or A   &   |- E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))   =>   |- sup(B, A, R) e. A
Theorem	supubi 5675	A supremum is an upper bound.
|- R Or A   &   |- E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))   =>   |- (C e. B -> -. sup(B, A, R)RC)
Theorem	suplubi 5676	A supremum is the least upper bound.
|- R Or A   &   |- E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))   =>   |- ((C e. A /\ CRsup(B, A, R)) -> E.z e. B CRz)
Theorem	supnubi 5677	An upper bound is not less than the supremum.
|- R Or A   &   |- E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz))   =>   |- ((C e. A /\ A.z e. B -. CRz) -> -. CRsup(B, A, R))
Theorem	supmaxlem 5678	A set that contains a greatest element satisfies the antecedent in supremum theorems. This allows sup(A, B, R) to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.)
|- ((C e. A /\ C e. B /\ A.z e. B -. CRz) -> E.x e. A (A.y e. B -. xRy /\ A.y e. A (yRx -> E.z e. B yRz)))
Theorem	supmax 5679	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.)
|- R Or A   =>   |- ((C e. A /\ C e. B /\ A.y e. B -. CRy) -> sup(B, A, R) = C)
Theorem	suppr 5680	The supremum of a pair.
|- R Or A   =>   |- ((B e. A /\ C e. A) -> sup({B, C}, A, R) = if(CRB, B, C))
Theorem	supsn 5681	The supremum of a singleton.
|- R Or A   =>   |- (B e. A -> sup({B}, A, R) = B)
Theorem	supsnALT 5682	The supremum of a singleton. This version of supsn 5681 is proved directly.
|- R Or A   =>   |- (B e. A -> sup({B}, A, R) = B)
Ordinal isomorphism, Hartog's theorem
Theorem	ordiso 5683	Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.)
|- ((A e. On /\ B e. On) -> (A = B <-> E.f f Isom _E , _E (A, B)))
Theorem	ordtypelem1 5684	Lemma for ordtype 5691.
Theorem	ordtypelem2 5685	Lemma for ordtype 5691.
Theorem	ordtypelem3 5686	Lemma for ordtype 5691.
Theorem	ordtypelem4 5687	Lemma for ordtype 5691. There is a smallest ordinal number whose image has no upper bounds in A.
Theorem	ordtypelem5 5688	Lemma for ordtype 5691. Establish injectivity of F when restricted to the ordinal number of ordtypelem4 5687.
Theorem	ordtypelem6 5689	Lemma for ordtype 5691. If z is not in (F"x), m is in (F"x), and every element of (F"x) which is less than m is also less than z, m is also less than z.
Theorem	ordtypelem7 5690	Lemma for ordtype 5691. F maps some ordinal isomorphically onto A.
Theorem	ordtype 5691	For any well-ordered set, there is an isomorphic ordinal number called its order type. (Contributed by Jeff Hankins, 17-Oct-2009.)
|- ((A e. B /\ R We A) -> E!x e. On E.f f Isom _E , R (x, A))
Theorem	hartoglem 5692	Lemma for hartog 5693.
Theorem	hartog 5693	Associate every set with an ordinal number known as its Hartog number. Apparently, this theorem has some sort of topological significance. I guess you would have to ask someone like Jeff Madsen or Dr. James Conant. Notice the similarity of this theorem to ondomon 6008, but unlike that theorem, this one does not require the Axiom of Choice ax-ac 5906. (Contributed by Jeff Hankins, 22-Oct-2009.)
|- H = {x e. On | x ~<_ A}   =>   |- (A e. B -> H e. On)
Theorem	onsdom 5694	Any ordinal number is strictly dominated by some other ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.)
|- (A e. On -> E.x e. On A ~< x)
ZF Set Theory - add the Axiom of Regularity
Introduce the Axiom of Regularity
Axiom	ax-reg 5695	Axiom of Regularity. An axiom of Zermelo-Fraenkel set theory. Also called the Axiom of Foundation. A rather non-intuitive axiom that denies more than it asserts, it states (in the form of zfreg 5698) that every non-empty set contains a set disjoint from itself. One consequence is that it denies the existence of a set containing itself (elirrv 5700). A stronger version that works for proper classes is proved as zfregs 5754.
|- (E.y y e. x -> E.y(y e. x /\ A.z(z e. y -> -. z e. x)))
Theorem	axreg 5696	Axiom of Regularity expressed more compactly.
|- (x e. y -> E.x(x e. y /\ A.z(z e. x -> -. z e. y)))
Theorem	zfregcl 5697	The Axiom of Regularity with class variables.
|- A e. _V   =>   |- (E.x x e. A -> E.x e. A A.y e. x -. y e. A)
Theorem	zfreg 5698	The Axiom of Regularity using abbreviations. Axiom 6 of [TakeutiZaring] p. 21. This is called the "weak form." There is also a "strong form," not requiring that A be a set, that can be proved with more difficulty (see zfregs 5754).
|- A e. _V   =>   |- (A =/= (/) -> E.x e. A (x i^i A) = (/))
Theorem	zfreg2 5699	The Axiom of Regularity using abbreviations. This form with the intersection arguments commuted (compared to zfreg 5698) is formally more convenient for us in some cases. Axiom Reg of [BellMachover] p. 480.
|- A e. _V   =>   |- (A =/= (/) -> E.x e. A (A i^i x) = (/))
Theorem	elirrv 5700	The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 5706 and efrirr 3637, but this proof is direct from the Axiom of Regularity.)
|- -. x e. x