mmtheorems56 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5501-5600 - Page 56 of 175

Type	Label	Description
Statement
Theorem	xpdom2 5501	Dominance law for cross product. Proposition 10.33(2) of [TakeutiZaring] p. 92.
|- B e. _V   &   |- C e. _V   =>   |- (A ~<_ B -> (C X. A) ~<_ (C X. B))
Theorem	xpdom1 5502	Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149.
|- B e. _V   &   |- C e. _V   =>   |- (A ~<_ B -> (A X. C) ~<_ (B X. C))
Theorem	xpdom1g 5503	Dominance law for cross product. Theorem 6L(c) of [Enderton] p. 149.
|- ((B e. R /\ C e. S /\ A ~<_ B) -> (A X. C) ~<_ (B X. C))
Theorem	xpdom3 5504	A set is dominated by its cross product with a non-empty set. Exercise 6 of [Suppes] p. 98.
|- A e. _V   =>   |- (B =/= (/) -> A ~<_ (A X. B))
Theorem	pw2en 5505	The power set of a set is equinumerous to set exponentiation with a base of ordinal 2. Proposition 10.44 of [TakeutiZaring] p. 96. (The proof seems excessively long. An attempt to find a shorter one is on my to-do list.)
|- A e. _V   =>   |- ~PA ~~ (2o ^m A)
Theorem	ac6sfilem1 5506	Lemma for ac6sfi 5509.
Theorem	ac6sfilem2 5507	Lemma for ac6sfi 5509.
Theorem	ac6sfilem3 5508	Lemma for ac6sfi 5509.
Theorem	ac6sfi 5509	A version of ac6s 5918 for finite sets. (Contributed by Jeffrey Hankins, 26-Jun-2009.)
|- (y = (f` x) -> (ph <-> ps))   =>   |- ((A e. Fin /\ A.x e. A E.y e. B ph) -> E.f(f:A-->B /\ A.x e. A ps))
Schroeder-Bernstein Theorem
Theorem	sbthlem1 5510	Lemma for sbth 5520.
Theorem	sbthlem2 5511	Lemma for sbth 5520.
Theorem	sbthlem3 5512	Lemma for sbth 5520.
Theorem	sbthlem4 5513	Lemma for sbth 5520.
Theorem	sbthlem5 5514	Lemma for sbth 5520.
Theorem	sbthlem6 5515	Lemma for sbth 5520.
Theorem	sbthlem7 5516	Lemma for sbth 5520.
Theorem	sbthlem8 5517	Lemma for sbth 5520.
Theorem	sbthlem9 5518	Lemma for sbth 5520.
Theorem	sbthlem10 5519	Lemma for sbth 5520.
Theorem	sbth 5520	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 5510 through sbthlem10 5519; this final piece mainly changes bound variables to eliminate the hypotheses of sbthlem10 5519. We follow closely the proof in Suppes, which you should consult to understand our proof at a higher level. Note that Suppes' proof, which is credited to J. M. Whitaker, does not require the Axiom of Infinity.
|- ((A ~<_ B /\ B ~<_ A) -> A ~~ B)
Theorem	sbthbg 5521	Schroeder-Bernstein Theorem and its converse.
|- (B e. C -> ((A ~<_ B /\ B ~<_ A) <-> A ~~ B))
Theorem	sbthcl 5522	Schroeder-Bernstein Theorem in class form.
|- ~~ = ( ~<_ i^i `' ~<_ )
Theorem	dfsdom2 5523	Alternate definition of strict dominance. Compare Definition 3 of [Suppes] p. 97.
|- ~< = ( ~<_ \ `' ~<_ )
Theorem	brsdom2 5524	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 5525	Strict dominance is not symmetric. Theorem 21(ii) of [Suppes] p. 97.
|- (A ~< B -> -. B ~< A)
Theorem	domnsym 5526	Theorem 22(i) of [Suppes] p. 97.
|- (A ~<_ B -> -. B ~< A)
Theorem	0dom 5527	Any set dominates the empty set.
|- (/) ~<_ A
Theorem	dom0 5528	A set dominated by the empty set is empty.
|- (A ~<_ (/) <-> A = (/))
Theorem	0sdomg 5529	A set strictly dominates the empty set iff it is not empty.
|- (A e. B -> ((/) ~< A <-> A =/= (/)))
Theorem	0sdom 5530	A set strictly dominates the empty set iff it is not empty.
|- A e. _V   =>   |- ((/) ~< A <-> A =/= (/))
Theorem	sdom0 5531	The empty set does not strictly dominate any set.
|- -. A ~< (/)
Theorem	sdomdomtr 5532	Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97.
|- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))
Theorem	sdomentr 5533	Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98.
|- (C e. D -> ((A ~< B /\ B ~~ C) -> A ~< C))
Theorem	ensdomtr 5534	Transitivity of equinumerosity and strict dominance.
|- ((A ~~ B /\ B ~< C) -> A ~< C)
Theorem	sdomirr 5535	Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97.
|- -. A ~< A
Theorem	sdomex 5536	Technical lemma for simplifying proofs involving strict dominance.
|- (A ~< B -> (A e. _V /\ B e. _V))
Theorem	sdomtr 5537	Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97.
|- ((A ~< B /\ B ~< C) -> A ~< C)
Theorem	sdomn2lp 5538	Strict dominance has no 2-cycle loops.
|- -. (A ~< B /\ B ~< A)
Theorem	domsdomtr 5539	Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97.
|- ((A ~<_ B /\ B ~< C) -> A ~< C)
Theorem	enen1 5540	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (A ~~ C <-> B ~~ C))
Theorem	enen2 5541	Equality-like theorem for equinumerosity.
|- ((B e. D /\ A ~~ B) -> (C ~~ A <-> C ~~ B))
Theorem	domen1 5542	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))
Theorem	domen2 5543	Equality-like theorem for equinumerosity and dominance.
|- ((B e. D /\ A ~~ B) -> (C ~<_ A <-> C ~<_ B))
Theorem	sdomen1 5544	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (A ~< C <-> B ~< C))
Theorem	sdomen2 5545	Equality-like theorem for equinumerosity and strict dominance.
|- ((B e. D /\ A ~~ B) -> (C ~< A <-> C ~< B))
Theorem	domtriord 5546	Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
|- ((A e. On /\ B e. On) -> (A ~<_ B <-> -. B ~< A))
Theorem	fodomr 5547	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 5548	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 5112.
|- A e. _V   =>   |- A ~< ~PA
Theorem	canth2g 5549	Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97.
|- (A e. B -> A ~< ~PA)
Theorem	pwuninel 5550	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.
|- -. ~PU.A e. A
Theorem	2pwuninel 5551	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~PU.A e. A
Theorem	pwne 5552	No set equals its power set.
|- (A e. B -> ~PA =/= A)
Theorem	2pwne 5553	No set equals the power set of its power set.
|- (A e. B -> ~P~PA =/= A)
Partial functions and restricted iota
Syntax	cund 5554	Extend class notation with undefined value function.
class Undef
Syntax	crio 5555	Extend class notation with restricted description binder.
class (iota_x e. Aph)
Definition	df-undef 5556	Define the undefined value function, whose value at set s is guaranteed not to be a member of s (see pwuninel 5550).
|- Undef = (s e. _V |-> ~PU.s)
Theorem	undefval 5557	Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 5559 instead.
|- S e. _V   =>   |- (Undef` S) = ~PU.S
Theorem	undefnel2 5558	The undefined value generated from a set is not a member of the set.
|- S e. _V   =>   |- -. (Undef` S) e. S
Theorem	undefnel 5559	The undefined value generated from a set is not a member of the set.
|- S e. _V   =>   |- (Undef` S) e/ S
Definition	df-riota 5560	Define restricted description binder. In case it doesn't exist, we return a set which is not a member of the domain of discourse A.
|- (iota_x e. Aph) = if(E!x e. A ph, (iotax(x e. A /\ ph)), (Undef` A))
Theorem	riotaeqdv 5561	Formula-building deduction rule for iota.
|- (ph -> A = B)   =>   |- (ph -> (iota_x e. Aps) = (iota_x e. Bps))
Theorem	riotabidv 5562	Formula-building deduction rule for restricted iota.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (iota_x e. Aps) = (iota_x e. Ach))
Theorem	riotaeqbidv 5563	Equality deduction for restricted universal quantifier.
|- (ph -> A = B)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> (iota_x e. Aps) = (iota_x e. Bch))
Theorem	riotaex 5564	Restricted iota is a set.
|- (iota_x e. Aps) e. _V
Theorem	riotav 5565	An iota restricted to the universe is unrestricted.
|- (iota_x e. _Vph) = (iotaxph)
Theorem	riotaiota 5566	Restricted iota in terms of iota.
|- (E!x e. A ph -> (iota_x e. Aph) = (iotax(x e. A /\ ph)))
Theorem	riotaprc 5567	For proper classes, restricted and unrestricted iota are the same.
|- (-. A e. _V -> (iota_x e. Aph) = (iotax(x e. A /\ ph)))
Theorem	riotauni 5568	Restricted iota in terms of class union.
|- (E!x e. A ph -> (iota_x e. Aph) = U.{x e. A | ph})
Theorem	hbriota1 5569	The abstraction variable in a restricted iota descriptor isn't free.
|- (y e. (iota_x e. Aph) -> A.x y e. (iota_x e. Aph))
Theorem	hbriota 5570	A variable not free in a wff remains so in a restricted iota descriptor.
|- (ph -> A.xph)   &   |- (z e. A -> A.x z e. A)   =>   |- (z e. (iota_y e. Aph) -> A.x z e. (iota_y e. Aph))
Theorem	riotacl 5571	Closure of restricted iota.
|- (E!x e. A ph -> (iota_x e. Aph) e. A)
Theorem	riotaund 5572	Restricted iota equals the undefined value of its domain of discourse A when not meaningful.
|- (-. E!x e. A ph -> (iota_x e. Aph) = (Undef` A))
Theorem	riotaclb 5573	Closure of restricted iota.
|- A e. _V   =>   |- (E!x e. A ph <-> (iota_x e. Aph) e. A)
Theorem	riotaundb 5574	Restricted iota equals the undefined value of its domain of discourse A when not meaningful.
|- A e. _V   =>   |- (-. E!x e. A ph <-> (iota_x e. Aph) = (Undef` A))
Theorem	riotabidva 5575	Equivalent wff's yield equal restricted class abstractions (deduction rule). (Th. rabbidva 2286 analog.)
|- ((ph /\ x e. A) -> (ps <-> ch))   =>   |- (ph -> (iota_x e. Aps) = (iota_x e. Ach))
Theorem	riotabiia 5576	Equivalent wff's yield equal restricted iotas (inference rule). (Th. rabbiia 2285 analog.)
|- (x e. A -> (ph <-> ps))   =>   |- (iota_x e. Aph) = (iota_x e. Aps)
Theorem	riota4 5577	Substitution law for descriptions. Compare reuuni4 3813.
|- (E!x e. A ph -> [(iota_x e. Aph) / x]ph)
Theorem	riotacl2 5578	Membership law for "the unique element in A such that ph." (Th. reucl2 3814 analog.)
|- (E!x e. A ph -> (iota_x e. Aph) e. {x e. A | ph})
Theorem	riota2f 5579	This theorem shows a condition that allows us to represent a descriptor with a class expression B. The second hypothesis is a weakened bound variable condition that allows hbsbc1g 2461 to be used. Compare reuuni2f 3810.
|- (y e. B -> A.x y e. B)   &   |- (B e. A -> (ps -> A.xps))   &   |- (x = B -> (ph <-> ps))   =>   |- ((B e. A /\ E!x e. A ph) -> (ps <-> (iota_x e. Aph) = B))
Theorem	riota5 5580	A method for computing restricted iota.
|- ((ph /\ B e. A /\ x e. A) -> (ps <-> x = B))   =>   |- ((ph /\ B e. A) -> (iota_x e. Aps) = B)
Theorem	riotaxfrd 5581	Change the variable x in the expression for "the unique A such that ph " to another variable y contained in expression B. Use reuhypd 3848 to eliminate the last hypothesis. (Th. reuunixfr 3850 analog.)
|- (z e. C -> A.y z e. C)   &   |- ((ph /\ y e. A) -> B e. A)   &   |- ((ph /\ (iota_y e. Ach) e. A) -> C e. A)   &   |- (x = B -> (ps <-> ch))   &   |- (y = (iota_y e. Ach) -> B = C)   &   |- ((ph /\ x e. A) -> E!y e. A x = B)   =>   |- ((ph /\ E!x e. A ps) -> (iota_x e. Aps) = C)
Equinumerosity (cont.)
Theorem	xpen 5582	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 5583	Lemma for mapen 5585.
Theorem	mapenlem2 5584	Lemma for mapen 5585.
Theorem	mapen 5585	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 5586	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 5587	Lemma for mapdom2 5588.
Theorem	mapdom2 5588	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 5589	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 5590	Lemma for xpmapen 5595.
Theorem	xpmapenlem2 5591	Lemma for xpmapen 5595.
Theorem	xpmapenlem3 5592	Lemma for xpmapen 5595.
Theorem	xpmapenlem4 5593	Lemma for xpmapen 5595.
Theorem	xpmapenlem5 5594	Lemma for xpmapen 5595.
Theorem	xpmapen 5595	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 5596	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 5597	If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87.
|- B e. _V   =>   |- (A ~~ B -> ~PA ~~ ~PB)
Theorem	ssenen 5598	Equinumerosity of equinumerous subsets of a set.
|- A e. _V   &   |- B e. _V   =>   |- (A ~~ B -> {x | (x C_ A /\ x ~~ C)} ~~ {x | (x C_ B /\ x ~~ C)})
Theorem	limenpsi 5599	A limit ordinal is equinumerous to a proper subset of itself.
|- Lim A   =>   |- (A e. B -> A ~~ (A \ {(/)}))
Theorem	limensuci 5600	A limit ordinal is equinumerous to its successor.
|- Lim A   =>   |- (A e. B -> A ~~ suc A)