mmtheorems47 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4601-4700 - Page 47 of 123

Type	Label	Description
Statement
Theorem	sbthlem10 4601	Lemma for sbth 4602.
Theorem	sbth 4602	Schroeder-Bernstein Theorem. Theorem 18 of [Suppes] p. 95. This theorem states that if set A is smaller (has lower cardinality) than B and vice-versa, then A and B are equinumerous (have the same cardinality). The interesting thing is that this can be proved without invoking the Axiom of Choice, as we do here, but the proof as you can see is quite difficult. (The theorem can be proved more easily if we allow AC.) The main proof consists of lemmas sbthlem1 4592 through sbthlem10 4601; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 4601. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level.
|- ((A ~<_ B /\ B ~<_ A) -> A ~~ B)
Theorem	sbthbg 4603	Schroeder-Bernstein Theorem and its converse.
|- (B e. C -> ((A ~<_ B /\ B ~<_ A) <-> A ~~ B))
Theorem	sbthcl 4604	Schroeder-Bernstein Theorem in class form.
|- ~~ = ( ~<_ i^i `' ~<_ )
Theorem	dfsdom2 4605	Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97.
|- ~< = ( ~<_ \ `' ~<_ )
Theorem	brsdom2 4606	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 4607	Strict dominance is not symmetric. Theorem 21(ii) of [Suppes] p. 97.
|- (A ~< B -> -. B ~< A)
Theorem	domnsym 4608	Theorem 22(i) of [Suppes] p. 97.
|- (A ~<_ B -> -. B ~< A)
Theorem	0dom 4609	Any set dominates the empty set.
|- (/) ~<_ A
Theorem	dom0 4610	A set dominated by the empty set is empty.
|- (A ~<_ (/) <-> A = (/))
Theorem	0sdomg 4611	A set strictly dominates the empty set iff it is not empty.
|- (A e. B -> ((/) ~< A <-> A =/= (/)))
Theorem	0sdom 4612	A set strictly dominates the empty set iff it is not empty.
|- A e. V   =>   |- ((/) ~< A <-> A =/= (/))
Theorem	sdom0 4613	The empty set does not strictly dominate any set.
|- -. A ~< (/)
Theorem	sdomdomtr 4614	Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97.
|- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))
Theorem	sdomentr 4615	Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98.
|- (C e. D -> ((A ~< B /\ B ~~ C) -> A ~< C))
Theorem	ensdomtr 4616	Transitivity of equinumerosity and strict dominance.
|- ((A ~~ B /\ B ~< C) -> A ~< C)
Theorem	sdomirr 4617	Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97.
|- -. A ~< A
Theorem	sdomex 4618	Technical lemma for simplifying proofs involving strict dominance.
|- (A ~< B -> (A e. V /\ B e. V))
Theorem	sdomtr 4619	Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97.
|- ((A ~< B /\ B ~< C) -> A ~< C)
Theorem	sdomn2lp 4620	Strict dominance has no 2-cycle loops.
|- -. (A ~< B /\ B ~< A)
Theorem	domsdomtr 4621	Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97.
|- ((A ~<_ B /\ B ~< C) -> A ~< C)
Theorem	enen1 4622	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (A ~~ C <-> B ~~ C))
Theorem	enen2 4623	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (C ~~ A <-> C ~~ B))
Theorem	domen1 4624	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))
Theorem	domen2 4625	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (C ~<_ A <-> C ~<_ B))
Theorem	sdomen1 4626	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (A ~< C <-> B ~< C))
Theorem	sdomen2 4627	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (C ~< A <-> C ~< B))
Theorem	fodomr 4628	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 4629	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 4205.
|- A e. V   =>   |- A ~< P~A
Theorem	canth2g 4630	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 4631	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 4632	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 4633	No set equals its power set.
|- (A e. B -> P~A =/= A)
Theorem	2pwne 4634	No set equals the power set of its power set.
|- (A e. B -> P~P~A =/= A)
Theorem	xpen 4635	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 4636	Lemma for mapen 4638.
Theorem	mapenlem2 4637	Lemma for mapen 4638.
Theorem	mapen 4638	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 4639	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 4640	Lemma for mapdom2 4641.
Theorem	mapdom2 4641	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 4642	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 4643	Lemma for xpmapen 4648.
Theorem	xpmapenlem2 4644	Lemma for xpmapen 4648.
Theorem	xpmapenlem3 4645	Lemma for xpmapen 4648.
Theorem	xpmapenlem4 4646	Lemma for xpmapen 4648.
Theorem	xpmapenlem5 4647	Lemma for xpmapen 4648.
Theorem	xpmapen 4648	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 4649	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 4650	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 4651	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 4652	A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
Theorem	limensuci 4653	A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)
Theorem	limensuc 4654	A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
Pigeonhole Principle
Theorem	phplem1 4655	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 4656	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
Theorem	phplem3 4657	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
Theorem	phplem4 4658	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
Theorem	nneneq 4659	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 4660	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 4655 through phplem4 4658, nneneq 4659, and this final piece of the proof.
|- ((A e. om /\ B (. A) -> -. A ~~ B)
Theorem	php2 4661	Corollary of Pigeonhole Principle.
|- ((A e. om /\ B (. A) -> B ~< A)
Theorem	php3 4662	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 4663	Corollary of the Pigeonhole Principle php 4660: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
Theorem	php5 4664	Corollary of the Pigeonhole Principle php 4660: 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 4665	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 4666	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 4667	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 4668	Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A (. B))
Theorem	omsucdom 4669	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 4670	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 4992.
|- ((A e. om /\ B e. C) -> (suc A ~<_ B -> A ~< B))
Theorem	0sdom1dom 4671	Strict dominance over zero is the same as dominance over one.
|- A e. V   =>   |- ((/) ~< A <-> 1o ~<_ A)
Theorem	1sdom2 4672	Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
Theorem	finsucdom 4673	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 4674	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 4675	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. om e. Fin
Theorem	omsdomnn 4676	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 4677	Omega strictly dominates a finite set. See comment in omsdomnn 4676.
|- (A e. Fin -> (A ~<_ om /\ -. om ~~ A))
Theorem	infsdomnn 4678	An infinite set strictly dominates a natural number.
|- A e. V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
Theorem	infn0 4679	An infinite set is not empty.
|- A e. V   =>   |- (om ~<_ A -> A =/= (/))
Theorem	enfi 4680	Equinmerous sets have the same finiteness.
|- ((B e. C /\ A ~~ B) -> (A e. Fin <-> B e. Fin))
Theorem	pssnn 4681	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 4682	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> B e. Fin)
Theorem	ssfi 4683	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((A e. Fin /\ B (_ A) -> B e. Fin)
Theorem	domfi 4684	A set dominated by a finite set is finite.
|- ((A e. Fin /\ B ~<_ A) -> B e. Fin)
Theorem	xpfi 4685	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 4686	Lemma for unbnn 4690. 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 4687	Lemma for unbnn 4690. The value of the function F belongs to the unbounded set of natural numbers A.
Theorem	unblem3 4688	Lemma for unbnn 4690. The value of the function F is less than its value at a successor.
Theorem	unblem4 4689	Lemma for unbnn 4690. The function F maps the set of natural numbers one-to-one to the set of unbounded natural numbers A.
Theorem	unbnn 4690	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 4785 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 4691	Version of unbnn 4690 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 4692	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 4693	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 4694	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem2 4695	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem3 4696	Lemma for proving that the union of two finite sets is finite.
Theorem	unfi 4697	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 4698	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. This version of unfi 4697 is useful only if we assume the Axiom of Infinity (see comments in fin2inf 4693).
|- ((A ~< om /\ B ~< om) -> (A u. B) ~< om)
Theorem	infcntss 4699	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 4700	An unordered pair is finite.
|- {A, B} e. Fin