mmtheorems45 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4401-4500 - Page 45 of 105

Type	Label	Description
Statement
Theorem	0sdom 4401	A set strictly dominates the empty set iff it is not empty.
|- A e. V   =>   |- ((/) ~< A <-> A =/= (/))
Theorem	sdom0 4402	The empty set does not strictly dominate any set.
|- -. A ~< (/)
Theorem	sdomdomtr 4403	Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97.
|- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))
Theorem	sdomentr 4404	Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98.
|- (C e. D -> ((A ~< B /\ B ~~ C) -> A ~< C))
Theorem	ensdomtr 4405	Transitivity of equinumerosity and strict dominance.
|- ((A ~~ B /\ B ~< C) -> A ~< C)
Theorem	sdomirr 4406	Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97.
|- -. A ~< A
Theorem	sdomex 4407	Technical lemma for simplifying proofs involving strict dominance.
|- (A ~< B -> (A e. V /\ B e. V))
Theorem	sdomtr 4408	Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97.
|- ((A ~< B /\ B ~< C) -> A ~< C)
Theorem	sdomn2lp 4409	Strict dominance has no 2-cycle loops.
|- -. (A ~< B /\ B ~< A)
Theorem	domsdomtr 4410	Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97.
|- ((A ~<_ B /\ B ~< C) -> A ~< C)
Theorem	enen1 4411	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (A ~~ C <-> B ~~ C))
Theorem	enen2 4412	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (C ~~ A <-> C ~~ B))
Theorem	domen1 4413	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))
Theorem	domen2 4414	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (C ~<_ A <-> C ~<_ B))
Theorem	sdomen1 4415	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (A ~< C <-> B ~< C))
Theorem	sdomen2 4416	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (C ~< A <-> C ~< B))
Theorem	fodomr 4417	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 4418	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 3846.
|- A e. V   =>   |- A ~< P~A
Theorem	canth2g 4419	Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97.
|- (A e. B -> A ~< P~A)
Theorem	xpen 4420	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 4421	Lemma for mapen 4423.
Theorem	mapenlem2 4422	Lemma for mapen 4423.
Theorem	mapen 4423	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 4424	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 4425	Lemma for mapdom2 4426.
Theorem	mapdom2 4426	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 4427	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 4428	Lemma for xpmapen 4433.
Theorem	xpmapenlem2 4429	Lemma for xpmapen 4433.
Theorem	xpmapenlem3 4430	Lemma for xpmapen 4433.
Theorem	xpmapenlem4 4431	Lemma for xpmapen 4433.
Theorem	xpmapenlem5 4432	Lemma for xpmapen 4433.
Theorem	xpmapen 4433	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 4434	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 4435	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 4436	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 4437	A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
Theorem	limensuci 4438	A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)
Theorem	limensuc 4439	A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
Pigeonhole Principle
Theorem	phplem1 4440	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 4441	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
Theorem	phplem3 4442	Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
Theorem	phplem4 4443	Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
Theorem	nneneq 4444	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 4445	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 4440 through phplem4 4443, nneneq 4444, and this final piece of the proof.
|- ((A e. om /\ B (. A) -> -. A ~~ B)
Theorem	php2 4446	Corollary of Pigeonhole Principle.
|- ((A e. om /\ B (. A) -> B ~< A)
Theorem	php3 4447	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. (The expression E.x e. omA ~~ x is the definition of "finite," and "infinite" is defined as "not finite.")
|- ((E.x e. om A ~~ x /\ B (. A) -> B ~< A)
Theorem	php4 4448	Corollary of the Pigeonhole Principle php 4445: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
Theorem	php5 4449	Corollary of the Pigeonhole Principle php 4445: 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 4450	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 4451	An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92.
|- (A e. On -> (E.x e. om A ~~ x <-> A e. om))
Theorem	nndomo 4452	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 4453	Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A (. B))
Theorem	omsucdom 4454	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 4455	Dominance of a set over a successor of a natural number implies strict dominance over the number. For the converse, see sucdom 4765.
|- ((A e. om /\ B e. C) -> (suc A ~<_ B -> A ~< B))
Theorem	0sdom1dom 4456	Strict dominance over zero is the same as dominance over one.
|- A e. V   =>   |- ((/) ~< A <-> 1o ~<_ A)
Theorem	1sdom2 4457	Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
Theorem	finsucdom 4458	Strict dominance of a finite set over a natural number is the same as dominance over its successor.
|- ((A e. om /\ E.x e. om B ~~ x) -> (A ~< B <-> suc A ~<_ B))
Theorem	pssinf 4459	A set equinumerous to a proper subset of itself is infinite. Corollary 6D(a) of [Enderton] p. 136.
|- ((A (. B /\ A ~~ B) -> -. E.x e. om B ~~ x)
Theorem	ominf 4460	The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. E.x e. om om ~~ x
Theorem	omsdomnn 4461	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 4462	Omega strictly dominates a finite set. See comment in omsdomnn 4461.
|- (E.x e. om A ~~ x -> (A ~<_ om /\ -. om ~~ A))
Theorem	infsdomnn 4463	An infinite set strictly dominates a natural number.
|- A e. V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
Theorem	infn0 4464	An infinite set is not empty.
|- A e. V   =>   |- (om ~<_ A -> A =/= (/))
Theorem	pssnn 4465	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	ssnn 4466	A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> E.x e. om B ~~ x)
Theorem	ssfi 4467	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((E.x e. om A ~~ x /\ B (_ A) -> E.x e. om B ~~ x)
Theorem	domfi 4468	A set dominated by a finite set is finite.
|- ((E.x e. om A ~~ x /\ B ~<_ A) -> E.x e. om B ~~ x)
Theorem	unblem1 4469	Lemma for unbnn 4473. 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 4470	Lemma for unbnn 4473. The value of the function F belongs to the unbounded set of natural numbers A.
Theorem	unblem3 4471	Lemma for unbnn 4473. The value of the function F is less than its value at a successor.
Theorem	unblem4 4472	Lemma for unbnn 4473. The function F maps the set of natural numbers one-to-one to the set of unbounded natural numbers A.
Theorem	unbnn 4473	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 unbnnt 4563 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 4474	Version of unbnn 4473 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 4475	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 -> E.x e. om A ~~ x)
Theorem	unfilem1 4476	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem2 4477	Lemma for proving that the union of two finite sets is finite.
Theorem	unfilem3 4478	Lemma for proving that the union of two finite sets is finite.
Theorem	fin2inf 4479	This theorem, which was proved without the Axiom of Infinity, is 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	unfi 4480	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144.
|- ((E.x e. om A ~~ x /\ E.x e. om B ~~ x) -> E.x e. om (A u. B) ~~ x)
Theorem	unfi2 4481	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 1