mmtheorems47 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4601-4700 - Page 47 of 111

Type	Label	Description
Statement
Theorem	dom0 4601	A set dominated by the empty set is empty.
|- (A ~<_ (/) <-> A = (/))
Theorem	0sdomg 4602	A set strictly dominates the empty set iff it is not empty.
|- (A e. B -> ((/) ~< A <-> A =/= (/)))
Theorem	0sdom 4603	A set strictly dominates the empty set iff it is not empty.
|- A e. V   =>   |- ((/) ~< A <-> A =/= (/))
Theorem	sdom0 4604	The empty set does not strictly dominate any set.
|- -. A ~< (/)
Theorem	sdomdomtr 4605	Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97.
|- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))
Theorem	sdomentr 4606	Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98.
|- (C e. D -> ((A ~< B /\ B ~~ C) -> A ~< C))
Theorem	ensdomtr 4607	Transitivity of equinumerosity and strict dominance.
|- ((A ~~ B /\ B ~< C) -> A ~< C)
Theorem	sdomirr 4608	Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97.
|- -. A ~< A
Theorem	sdomex 4609	Technical lemma for simplifying proofs involving strict dominance.
|- (A ~< B -> (A e. V /\ B e. V))
Theorem	sdomtr 4610	Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97.
|- ((A ~< B /\ B ~< C) -> A ~< C)
Theorem	sdomn2lp 4611	Strict dominance has no 2-cycle loops.
|- -. (A ~< B /\ B ~< A)
Theorem	domsdomtr 4612	Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97.
|- ((A ~<_ B /\ B ~< C) -> A ~< C)
Theorem	enen1 4613	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (A ~~ C <-> B ~~ C))
Theorem	enen2 4614	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (C ~~ A <-> C ~~ B))
Theorem	domen1 4615	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))
Theorem	domen2 4616	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (C ~<_ A <-> C ~<_ B))
Theorem	sdomen1 4617	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (A ~< C <-> B ~< C))
Theorem	sdomen2 4618	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (C ~< A <-> C ~< B))
Theorem	fodomr 4619	There exists a mapping from a set onto any (non-empty) set that it dominates.
|- ((A e. C /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
Theorem	canth2 4620	Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 4200.
|- A e. V   =>   |- A ~< P~A
Theorem	canth2g 4621	Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97.
|- (A e. B -> A ~< P~A)
Theorem	pwuninel 4622	The power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set.
|- -. P~U.A e. A
Theorem	2pwuninel 4623	The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set.
|- -. P~P~U.A e. A
Theorem	pwne 4624	No set equals its power set.
|- (A e. B -> P~A =/= A)
Theorem	2pwne 4625	No set equals the power set of its power set.
|- (A e. B -> P~P~A =/= A)
Theorem	xpen 4626	Equinumerosity law for cross product. Proposition 4.22(b) of [Mendelson] p. 254.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- ((A ~~ B /\ C ~~ D) -> (A X. C) ~~ (B X. D))
Theorem	mapenlem1 4627	Lemma for mapen 4629.
Theorem	mapenlem2 4628	Lemma for mapen 4629.
Theorem	mapen 4629	Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- D e. V   =>   |- ((A ~~ B /\ C ~~ D) -> (A ^m C) ~~ (B ^m D))
Theorem	mapdom1 4630	Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- (A ~<_ B -> (A ^m C) ~<_ (B ^m C))
Theorem	mapdom2lem 4631	Lemma for mapdom2 4632.
Theorem	mapdom2 4632	Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A ~<_ B /\ -. (A = (/) /\ C = (/))) -> (C ^m A) ~<_ (C ^m B))
Theorem	mapxpen 4633	Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A ^m B) ^m C) ~~ (A ^m (B X. C))
Theorem	xpmapenlem1 4634	Lemma for xpmapen 4639.
Theorem	xpmapenlem2 4635	Lemma for xpmapen 4639.
Theorem	xpmapenlem3 4636	Lemma for xpmapen 4639.
Theorem	xpmapenlem4 4637	Lemma for xpmapen 4639.
Theorem	xpmapenlem5 4638	Lemma for xpmapen 4639.
Theorem	xpmapen 4639	Equinumerosity law for set exponentiation of a cross product. Exercise 4.47 of [Mendelson] p. 255.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A X. B) ^m C) ~~ ((A ^m C) X. (B ^m C))
Theorem	mapunen 4640	Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255.
|- A e. V   &   |- B e. V   &   |- C e. V   =>   |- ((A i^i B) = (/) -> (C ^m (A u. B)) ~~ ((C ^m A) X. (C ^m B)))
Theorem	pwen 4641	If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87.
|- B e. V   =>   |- (A ~~ B -> P~A ~~ P~B)
Theorem	ssenen 4642	Equinumerosity of equinumerous subsets of a set.
|- A e. V   &   |- B e. V   =>   |- (A ~~ B -> {x | (x (_ A /\ x ~~ C)} ~~ {x | (x (_ B /\ x ~~ C)})
Theorem	limenpsi 4643	A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
Theorem	limensuci 4644	A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)
Theorem	limensuc 4645	A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
Pigeonhole Principle
Theorem	phplem1 4646	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 4647	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
Theorem	phplem3 4648	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
Theorem	phplem4 4649	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
Theorem	nneneq 4650	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 4651	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 4646 through phplem4 4649, nneneq 4650, and this final piece of the proof.
|- ((A e. om /\ B (. A) -> -. A ~~ B)
Theorem	php2 4652	Corollary of Pigeonhole Principle.
|- ((A e. om /\ B (. A) -> B ~< A)
Theorem	php3 4653	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 (. A) -> B ~< A)
Theorem	php4 4654	Corollary of the Pigeonhole Principle php 4651: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
Theorem	php5 4655	Corollary of the Pigeonhole Principle php 4651: 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 4656	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 4657	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 4658	Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146.
|- ((A e. om /\ B e. om) -> (A ~<_ B <-> A (_ B))
Theorem	nnsdomo 4659	Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A (. B))
Theorem	omsucdom 4660	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 4661	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 4983.
|- ((A e. om /\ B e. C) -> (suc A ~<_ B -> A ~< B))
Theorem	0sdom1dom 4662	Strict dominance over zero is the same as dominance over one.
|- A e. V   =>   |- ((/) ~< A <-> 1o ~<_ A)
Theorem	1sdom2 4663	Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
Theorem	finsucdom 4664	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 4665	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136.
|- ((A (. B /\ A ~~ B) -> -. B e. Fin)
Theorem	ominf 4666	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. om e. Fin
Theorem	omsdomnn 4667	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 4668	Omega strictly dominates a finite set. See comment in omsdomnn 4667.
|- (A e. Fin -> (A ~<_ om /\ -. om ~~ A))
Theorem	infsdomnn 4669	An infinite set strictly dominates a natural number.
|- A e. V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
Theorem	infn0 4670	An infinite set is not empty.
|- A e. V   =>   |- (om ~<_ A -> A =/= (/))
Theorem	enfi 4671	Equinmerous sets have the same finiteness.
|- ((B e. C /\ A ~~ B) -> (A e. Fin <-> B e. Fin))
Theorem	pssnn 4672	A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137.
|- ((A e. om /\ B (. A) -> E.x e. A B ~~ x)
Theorem	ssnnfi 4673	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> B e. Fin)
Theorem	ssfi 4674	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((A e. Fin /\ B (_ A) -> B e. Fin)
Theorem	domfi 4675	A set dominated by a finite set is finite.
|- ((A e. Fin /\ B ~<_ A) -> B e. Fin)
Theorem	xpfi 4676	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 4677	Lemma for unbnn 4681. 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 4678	Lemma for unbnn 4681. The value of the function F belongs to the unbounded set of natural numbers A.
Theorem	unblem3 4679	Lemma for unbnn 4681. The value of the function F is less than its value at a successor.
Theorem	unblem4 4680	Lemma for unbnn 4681. The function F maps the set of natural numbers one-to-one to the set of unbounded natural numbers A.
Theorem	unbnn 4681	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 4776 for a stronger version without the hypothesis.
|- A e. V   =>   |- ((A (_ om /\ A.x e. om E.y e. A x e. y) -> A ~~ om)
Theorem	unbnn2 4682	Version of unbnn 4681 that does not require a strict upper bound.
|- A e. V   =>   |- ((A (_ om /\ A.x e. om E.y e. A x (_ y) -> A ~~ om)
Theorem	isfinite2 4683	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 4684	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 4685	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem2 4686	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem3 4687	Lemma for proving that the union of two finite sets is finite.
Theorem	unfi 4688	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 4689	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 4688 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4684).
|- ((A ~< om /\ B ~< om) -> (A u. B) ~< om)
Theorem	infcntss 4690	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 (_ A /\ x ~~ om))
Theorem	prfi 4691	An unordered pair is finite.
|- {A, B} e. Fin
Theorem	unifi 4692	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 4693	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 4692 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4684).
|- ((A ~< om /\ A.x e. A x ~< om) -> U.A ~< om)
Theorem	fiint 4694	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 (_ A /\ x =/= (/) /\ x e. Fin) -> |^|x e. A))
Theorem	abfii1 4695	Two ways to express the collection of finite intersections of a set A.
|- |^|{x | (A (_ x /\ A.y e. x A.z e. x (y i^i z) e. x)} = |^|{x | (A (_ x /\ A.y((y (_ x /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))}
Theorem	abfii2 4696	Two ways to express the collection of finite intersections of a set A.
|- A e. V   =>   |- {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)}
Theorem	abfii3 4697	Two ways to express the collection of finite intersections of a set A.
|- A e. V   =>   |- |^|{x | (A (_ x /\ A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x)}
Theorem	abfii4 4698	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 (_ x /\ A.y((y (_ x /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))} = |^|{x | (A (_ x /\ A.y((y (_ A /\ y =/= (/) /\ y e. Fin) -> |^|y e. x))}
Theorem	abfii5 4699	Two ways to express the collection of finite intersections of a set A.
|- A e. V   =>   |- |^|{x | (A (_ x /\ A.y e. x A.z e. x (y i^i z) e. x)} = {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)}
Theorem	fodomfi 4700	An onto function implies dominance of domain over range, for finite sets. Unlike fodom 4935 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof.
|- ((A e. Fin /\ F:A-onto->B) -> B ~<_ A)