mmtheorems47 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4601-4700 - Page 47 of 122

Type	Label	Description
Statement
Theorem	sbthbg 4601	Schroeder-Bernstein Theorem and its converse.
|- (B e. C -> ((A ~<_ B /\ B ~<_ A) <-> A ~~ B))
Theorem	sbthcl 4602	Schroeder-Bernstein Theorem in class form.
|- ~~ = ( ~<_ i^i `' ~<_ )
Theorem	dfsdom2 4603	Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97.
|- ~< = ( ~<_ \ `' ~<_ )
Theorem	brsdom2 4604	Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97.
|- A e. V   &   |- B e. V   =>   |- (A ~< B <-> (A ~<_ B /\ -. B ~<_ A))
Theorem	sdomnsym 4605	Strict dominance is not symmetric. Theorem 21(ii) of [Suppes] p. 97.
|- (A ~< B -> -. B ~< A)
Theorem	domnsym 4606	Theorem 22(i) of [Suppes] p. 97.
|- (A ~<_ B -> -. B ~< A)
Theorem	0dom 4607	Any set dominates the empty set.
|- (/) ~<_ A
Theorem	dom0 4608	A set dominated by the empty set is empty.
|- (A ~<_ (/) <-> A = (/))
Theorem	0sdomg 4609	A set strictly dominates the empty set iff it is not empty.
|- (A e. B -> ((/) ~< A <-> A =/= (/)))
Theorem	0sdom 4610	A set strictly dominates the empty set iff it is not empty.
|- A e. V   =>   |- ((/) ~< A <-> A =/= (/))
Theorem	sdom0 4611	The empty set does not strictly dominate any set.
|- -. A ~< (/)
Theorem	sdomdomtr 4612	Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97.
|- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))
Theorem	sdomentr 4613	Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98.
|- (C e. D -> ((A ~< B /\ B ~~ C) -> A ~< C))
Theorem	ensdomtr 4614	Transitivity of equinumerosity and strict dominance.
|- ((A ~~ B /\ B ~< C) -> A ~< C)
Theorem	sdomirr 4615	Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97.
|- -. A ~< A
Theorem	sdomex 4616	Technical lemma for simplifying proofs involving strict dominance.
|- (A ~< B -> (A e. V /\ B e. V))
Theorem	sdomtr 4617	Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97.
|- ((A ~< B /\ B ~< C) -> A ~< C)
Theorem	sdomn2lp 4618	Strict dominance has no 2-cycle loops.
|- -. (A ~< B /\ B ~< A)
Theorem	domsdomtr 4619	Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97.
|- ((A ~<_ B /\ B ~< C) -> A ~< C)
Theorem	enen1 4620	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (A ~~ C <-> B ~~ C))
Theorem	enen2 4621	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (C ~~ A <-> C ~~ B))
Theorem	domen1 4622	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))
Theorem	domen2 4623	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (C ~<_ A <-> C ~<_ B))
Theorem	sdomen1 4624	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (A ~< C <-> B ~< C))
Theorem	sdomen2 4625	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (C ~< A <-> C ~< B))
Theorem	fodomr 4626	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 4627	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 4203.
|- A e. V   =>   |- A ~< P~A
Theorem	canth2g 4628	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 4629	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 4630	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 4631	No set equals its power set.
|- (A e. B -> P~A =/= A)
Theorem	2pwne 4632	No set equals the power set of its power set.
|- (A e. B -> P~P~A =/= A)
Theorem	xpen 4633	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 4634	Lemma for mapen 4636.
Theorem	mapenlem2 4635	Lemma for mapen 4636.
Theorem	mapen 4636	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 4637	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 4638	Lemma for mapdom2 4639.
Theorem	mapdom2 4639	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 4640	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 4641	Lemma for xpmapen 4646.
Theorem	xpmapenlem2 4642	Lemma for xpmapen 4646.
Theorem	xpmapenlem3 4643	Lemma for xpmapen 4646.
Theorem	xpmapenlem4 4644	Lemma for xpmapen 4646.
Theorem	xpmapenlem5 4645	Lemma for xpmapen 4646.
Theorem	xpmapen 4646	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 4647	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 4648	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 4649	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 4650	A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
Theorem	limensuci 4651	A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)
Theorem	limensuc 4652	A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
Pigeonhole Principle
Theorem	phplem1 4653	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 4654	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
Theorem	phplem3 4655	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
Theorem	phplem4 4656	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
Theorem	nneneq 4657	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 4658	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 4653 through phplem4 4656, nneneq 4657, and this final piece of the proof.
|- ((A e. om /\ B (. A) -> -. A ~~ B)
Theorem	php2 4659	Corollary of Pigeonhole Principle.
|- ((A e. om /\ B (. A) -> B ~< A)
Theorem	php3 4660	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 4661	Corollary of the Pigeonhole Principle php 4658: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
Theorem	php5 4662	Corollary of the Pigeonhole Principle php 4658: 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 4663	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 4664	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 4665	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 4666	Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A (. B))
Theorem	omsucdom 4667	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 4668	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 4990.
|- ((A e. om /\ B e. C) -> (suc A ~<_ B -> A ~< B))
Theorem	0sdom1dom 4669	Strict dominance over zero is the same as dominance over one.
|- A e. V   =>   |- ((/) ~< A <-> 1o ~<_ A)
Theorem	1sdom2 4670	Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
Theorem	finsucdom 4671	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 4672	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 4673	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. om e. Fin
Theorem	omsdomnn 4674	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 4675	Omega strictly dominates a finite set. See comment in omsdomnn 4674.
|- (A e. Fin -> (A ~<_ om /\ -. om ~~ A))
Theorem	infsdomnn 4676	An infinite set strictly dominates a natural number.
|- A e. V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
Theorem	infn0 4677	An infinite set is not empty.
|- A e. V   =>   |- (om ~<_ A -> A =/= (/))
Theorem	enfi 4678	Equinmerous sets have the same finiteness.
|- ((B e. C /\ A ~~ B) -> (A e. Fin <-> B e. Fin))
Theorem	pssnn 4679	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 4680	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> B e. Fin)
Theorem	ssfi 4681	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((A e. Fin /\ B (_ A) -> B e. Fin)
Theorem	domfi 4682	A set dominated by a finite set is finite.
|- ((A e. Fin /\ B ~<_ A) -> B e. Fin)
Theorem	xpfi 4683	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 4684	Lemma for unbnn 4688. 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 4685	Lemma for unbnn 4688. The value of the function F belongs to the unbounded set of natural numbers A.
Theorem	unblem3 4686	Lemma for unbnn 4688. The value of the function F is less than its value at a successor.
Theorem	unblem4 4687	Lemma for unbnn 4688. The function F maps the set of natural numbers one-to-one to the set of unbounded natural numbers A.
Theorem	unbnn 4688	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 4783 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 4689	Version of unbnn 4688 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 4690	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 4691	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 4692	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem2 4693	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem3 4694	Lemma for proving that the union of two finite sets is finite.
Theorem	unfi 4695	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 4696	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 4695 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4691).
|- ((A ~< om /\ B ~< om) -> (A u. B) ~< om)
Theorem	infcntss 4697	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 4698	An unordered pair is finite.
|- {A, B} e. Fin
Theorem	unifi 4699	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 4700	The finite union of finite sets is finite. Exercise 13 of [Enderton] p. 144. This version of unifi 4699 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4691).
|- ((A ~< om /\ A.x e. A x ~< om) -> U.A ~< om)