HomeHome Metamath Proof Explorer < Previous   Next >
Browser slow? Try the
Unicode version.

Jump to page: Contents + 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11098

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-9039)
  Hilbert Space Explorer  Hilbert Space Explorer
(9040-10625)
  User Sandboxes  User Sandboxes
(10626-11098)
 

Statement List for Metamath Proof Explorer - 4601-4700 - Page 47 of 111
TypeLabelDescription
Statement
 
Theoremdom0 4601 A set dominated by the empty set is empty.
|- (A ~<_ (/) <-> A = (/))
 
Theorem0sdomg 4602 A set strictly dominates the empty set iff it is not empty.
|- (A e. B -> ((/) ~< A <-> A =/= (/)))
 
Theorem0sdom 4603 A set strictly dominates the empty set iff it is not empty.
|- A e. V   =>   |- ((/) ~< A <-> A =/= (/))
 
Theoremsdom0 4604 The empty set does not strictly dominate any set.
|- -. A ~< (/)
 
Theoremsdomdomtr 4605 Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97.
|- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))
 
Theoremsdomentr 4606 Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98.
|- (C e. D -> ((A ~< B /\ B ~~ C) -> A ~< C))
 
Theoremensdomtr 4607 Transitivity of equinumerosity and strict dominance.
|- ((A ~~ B /\ B ~< C) -> A ~< C)
 
Theoremsdomirr 4608 Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97.
|- -. A ~< A
 
Theoremsdomex 4609 Technical lemma for simplifying proofs involving strict dominance.
|- (A ~< B -> (A e. V /\ B e. V))
 
Theoremsdomtr 4610 Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97.
|- ((A ~< B /\ B ~< C) -> A ~< C)
 
Theoremsdomn2lp 4611 Strict dominance has no 2-cycle loops.
|- -. (A ~< B /\ B ~< A)
 
Theoremdomsdomtr 4612 Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97.
|- ((A ~<_ B /\ B ~< C) -> A ~< C)
 
Theoremenen1 4613 Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (A ~~ C <-> B ~~ C))
 
Theoremenen2 4614 Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (C ~~ A <-> C ~~ B))
 
Theoremdomen1 4615 Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))
 
Theoremdomen2 4616 Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (C ~<_ A <-> C ~<_ B))
 
Theoremsdomen1 4617 Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (A ~< C <-> B ~< C))
 
Theoremsdomen2 4618 Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (C ~< A <-> C ~< B))
 
Theoremfodomr 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)
 
Theoremcanth2 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
 
Theoremcanth2g 4621 Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97.
|- (A e. B -> A ~< P~A)
 
Theorempwuninel 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
 
Theorem2pwuninel 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
 
Theorempwne 4624 No set equals its power set.
|- (A e. B -> P~A =/= A)
 
Theorem2pwne 4625 No set equals the power set of its power set.
|- (A e. B -> P~P~A =/= A)
 
Theoremxpen 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))
 
Theoremmapenlem1 4627 Lemma for mapen 4629.
 
Theoremmapenlem2 4628 Lemma for mapen 4629.
 
Theoremmapen 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))
 
Theoremmapdom1 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))
 
Theoremmapdom2lem 4631 Lemma for mapdom2 4632.
 
Theoremmapdom2 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))
 
Theoremmapxpen 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))
 
Theoremxpmapenlem1 4634 Lemma for xpmapen 4639.
 
Theoremxpmapenlem2 4635 Lemma for xpmapen 4639.
 
Theoremxpmapenlem3 4636 Lemma for xpmapen 4639.
 
Theoremxpmapenlem4 4637 Lemma for xpmapen 4639.
 
Theoremxpmapenlem5 4638 Lemma for xpmapen 4639.
 
Theoremxpmapen 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))
 
Theoremmapunen 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)))
 
Theorempwen 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)
 
Theoremssenen 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)})
 
Theoremlimenpsi 4643 A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
 
Theoremlimensuci 4644 A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)
 
Theoremlimensuc 4645 A limit ordinal is equinumerous to its successor.
|- ((A e. B /\ Lim A) -> A ~~ suc A)
 
Pigeonhole Principle
 
Theoremphplem1 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.
 
Theoremphplem2 4647 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.
 
Theoremphplem3 4648 Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.
 
Theoremphplem4 4649 Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers.
 
Theoremnneneq 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))
 
Theoremphp 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)
 
Theoremphp2 4652 Corollary of Pigeonhole Principle.
|- ((A e. om /\ B (. A) -> B ~< A)
 
Theoremphp3 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)
 
Theoremphp4 4654 Corollary of the Pigeonhole Principle php 4651: a natural number is strictly dominated by its successor.
|- (A e. om -> A ~< suc A)
 
Theoremphp5 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
 
Theoremonomeneq 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))
 
Theoremonfin 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))
 
Theoremnndomo 4658 Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146.
|- ((A e. om /\ B e. om) -> (A ~<_ B <-> A (_ B))
 
Theoremnnsdomo 4659 Cardinal ordering agrees with natural number ordering.
|- ((A e. om /\ B e. om) -> (A ~< B <-> A (. B))
 
Theoremomsucdom 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))
 
Theoremsucdomi 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))
 
Theorem0sdom1dom 4662 Strict dominance over zero is the same as dominance over one.
|- A e. V   =>   |- ((/) ~< A <-> 1o ~<_ A)
 
Theorem1sdom2 4663 Ordinal 1 is strictly dominated by ordinal 2.
|- 1o ~< 2o
 
Theoremfinsucdom 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))
 
Theorempssinf 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)
 
Theoremominf 4666 The set of natural numbers is infinite. Corollary 6D(b) of [Enderton] p. 136.
|- -. om e. Fin
 
Theoremomsdomnn 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))
 
Theoremisfinite1 4668 Omega strictly dominates a finite set. See comment in omsdomnn 4667.
|- (A e. Fin -> (A ~<_ om /\ -. om ~~ A))
 
Theoreminfsdomnn 4669 An infinite set strictly dominates a natural number.
|- A e. V   =>   |- ((om ~<_ A /\ B e. om) -> B ~< A)
 
Theoreminfn0 4670 An infinite set is not empty.
|- A e. V   =>   |- (om ~<_ A -> A =/= (/))
 
Theoremenfi 4671 Equinmerous sets have the same finiteness.
|- ((B e. C /\ A ~~ B) -> (A e. Fin <-> B e. Fin))
 
Theorempssnn 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)
 
Theoremssnnfi 4673 A subset of a natural number is finite.
|- ((A e. om /\ B (_ A) -> B e. Fin)
 
Theoremssfi 4674 A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138.
|- ((A e. Fin /\ B (_ A) -> B e. Fin)
 
Theoremdomfi 4675 A set dominated by a finite set is finite.
|- ((A e. Fin /\ B ~<_ A) -> B e. Fin)
 
Theoremxpfi 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))
 
Theoremunblem1 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.
 
Theoremunblem2 4678 Lemma for unbnn 4681. The value of the function F belongs to the unbounded set of natural numbers A.
 
Theoremunblem3 4679 Lemma for unbnn 4681. The value of the function F is less than its value at a successor.
 
Theoremunblem4 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.
 
Theoremunbnn 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)
 
Theoremunbnn2 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)
 
Theoremisfinite2 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)
 
Theoremfin2inf 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)
 
Theoremunfilem1 4685 Lemma for proving that the union of two finite sets is finite.
 
Theoremunfilem2 4686 Lemma for proving that the union of two finite sets is finite.
 
Theoremunfilem3 4687 Lemma for proving that the union of two finite sets is finite.
 
Theoremunfi 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)
 
Theoremunfi2 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)
 
Theoreminfcntss 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))
 
Theoremprfi 4691 An unordered pair is finite.
|- {A, B} e. Fin
 
Theoremunifi 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)
 
Theoremunifi2 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)
 
Theoremfiint 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))
 
Theoremabfii1 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))}
 
Theoremabfii2 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)}
 
Theoremabfii3 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)}
 
Theoremabfii4 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))}
 
Theoremabfii5 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)}
 
Theoremfodomfi 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)

MPE Home   Contents Copyright terms: Public domain < Previous  Next >