P. 89 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8801-8900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isf32lem12 8801*	Lemma for isfin3-2 8804. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- F = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }   =>    |- ( G e. V -> ( -. om ~<_* G -> G e. F ) )
Theorem	isfin32i 8802	One half of isfin3-2 8804. (Contributed by Mario Carneiro, 3-Jun-2015.)
|- ( A e. FinIII -> -. om ~<_* A )
Theorem	isf33lem 8803*	Lemma for isfin3-3 8805. (Contributed by Stefan O'Rear, 17-May-2015.)
|- FinIII = { g | A. a e. ( ~P g ^m om ) ( A. x e. om ( a ` suc x ) C_ ( a ` x ) -> |^| ran a e. ran a ) }
Theorem	isfin3-2 8804	Weakly Dedekind-infinite sets are exactly those which can be mapped onto om. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- ( A e. V -> ( A e. FinIII <-> -. om ~<_* A ) )
Theorem	isfin3-3 8805*	Weakly Dedekind-infinite sets are exactly those with an om-indexed descending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- ( A e. V -> ( A e. FinIII <-> A. f e. ( ~P A ^m om ) ( A. x e. om ( f ` suc x ) C_ ( f ` x ) -> |^| ran f e. ran f ) ) )
Theorem	fin33i 8806*	Inference from isfin3-3 8805. (This is actually a bit stronger than isfin3-3 8805 because it does not assume F is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)
|- ( ( A e. FinIII /\ F : om --> ~P A /\ A. x e. om ( F ` suc x ) C_ ( F ` x ) ) -> |^| ran F e. ran F )
Theorem	compsscnvlem 8807*	Lemma for compsscnv 8808. (Contributed by Mario Carneiro, 17-May-2015.)
|- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )
Theorem	compsscnv 8808*	Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- `' F = F
Theorem	isf34lem1 8809*	Lemma for isfin3-4 8819. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( ( A e. V /\ X C_ A ) -> ( F ` X ) = ( A \ X ) )
Theorem	isf34lem2 8810*	Lemma for isfin3-4 8819. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( A e. V -> F : ~P A --> ~P A )
Theorem	compssiso 8811*	Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )
Theorem	isf34lem3 8812*	Lemma for isfin3-4 8819. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )
Theorem	compss 8813*	Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( F " G ) = { y e. ~P A | ( A \ y ) e. G }
Theorem	isf34lem4 8814*	Lemma for isfin3-4 8819. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. X ) = |^| ( F " X ) )
Theorem	isf34lem5 8815*	Lemma for isfin3-4 8819. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) )
Theorem	isf34lem7 8816*	Lemma for isfin3-4 8819. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( ( A e. FinIII /\ G : om --> ~P A /\ A. y e. om ( G ` y ) C_ ( G ` suc y ) ) -> U. ran G e. ran G )
Theorem	isf34lem6 8817*	Lemma for isfin3-4 8819. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- F = ( x e. ~P A |-> ( A \ x ) )   =>    |- ( A e. V -> ( A e. FinIII <-> A. f e. ( ~P A ^m om ) ( A. y e. om ( f ` y ) C_ ( f ` suc y ) -> U. ran f e. ran f ) ) )
Theorem	fin34i 8818*	Inference from isfin3-4 8819. (Contributed by Mario Carneiro, 17-May-2015.)
|- ( ( A e. FinIII /\ G : om --> ~P A /\ A. x e. om ( G ` x ) C_ ( G ` suc x ) ) -> U. ran G e. ran G )
Theorem	isfin3-4 8819*	Weakly Dedekind-infinite sets are exactly those with an om-indexed ascending chain of subsets. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- ( A e. V -> ( A e. FinIII <-> A. f e. ( ~P A ^m om ) ( A. x e. om ( f ` x ) C_ ( f ` suc x ) -> U. ran f e. ran f ) ) )
Theorem	fin11a 8820	Every I-finite set is Ia-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin -> A e. FinIa )
Theorem	enfin1ai 8821	Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
|- ( A ~~ B -> ( A e. FinIa -> B e. FinIa ) )
Theorem	isfin1-2 8822	A set is finite in the usual sense iff the power set of its power set is Dedekind finite. (Contributed by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin <-> ~P ~P A e. FinIV )
Theorem	isfin1-3 8823	A set is I-finite iff every system of subsets contains a maximal subset. Definition I of [Levy58] p. 2. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- ( A e. V -> ( A e. Fin <-> `' [ C.] Fr ~P A ) )
Theorem	isfin1-4 8824	A set is I-finite iff every system of subsets contains a minimal subset. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. V -> ( A e. Fin <-> [ C.] Fr ~P A ) )
Theorem	dffin1-5 8825	Compact quantifier-free version of the standard definition df-fin 7584. (Contributed by Stefan O'Rear, 6-Jan-2015.)
|- Fin = ( ~~ " om )
Theorem	fin23 8826	Every II-finite set (every chain of subsets has a maximal element) is III-finite (has no denumerable collection of subsets). The proof here is the only one I could find, from http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm619.pdf p.94 (writeup by Tarski, credited to Kuratowski). Translated into English and modern notation, the proof proceeds as follows (variables renamed for uniqueness): Suppose for a contradiction that A is a set which is II-finite but not III-finite. For any countable sequence of distinct subsets T of A, we can form a decreasing sequence of nonempty subsets ( U ` T ) by taking finite intersections of initial segments of T while skipping over any element of T which would cause the intersection to be empty. By II-finiteness (as fin2i2 8755) this sequence contains its intersection, call it Y; since by induction every subset in the sequence U is nonempty, the intersection must be nonempty. Suppose that an element X of T has nonempty intersection with Y. Thus, said element has a nonempty intersection with the corresponding element of U, therefore it was used in the construction of U and all further elements of U are subsets of X, thus X contains the Y. That is, all elements of X either contain Y or are disjoint from it. Since there are only two cases, there must exist an infinite subset of T which uniformly either contain Y or are disjoint from it. In the former case we can create an infinite set by subtracting Y from each element. In either case, call the result Z; this is an infinite set of subsets of A, each of which is disjoint from Y and contained in the union of T; the union of Z is strictly contained in the union of T, because only the latter is a superset of the nonempty set Y. The preceding four steps may be iterated a countable number of times starting from the assumed denumerable set of subsets to produce a denumerable sequence B of the T sets from each stage. Great caution is required to avoid ax-dc 8883 here; in particular an effective version of the pigeonhole principle (for aleph-null pigeons and 2 holes) is required. Since a denumerable set of subsets is assumed to exist, we can conclude om e. _V without the axiom. This B sequence is strictly decreasing, thus it has no minimum, contradicting the first assumption. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- ( A e. FinII -> A e. FinIII )
Theorem	fin34 8827	Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
|- ( A e. FinIII -> A e. FinIV )
Theorem	isfin5-2 8828	Alternate definition of V-finite which emphasizes the idempotent behavior of V-infinite sets. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. V -> ( A e. FinV <-> -. ( A =/= (/) /\ A ~~ ( A +c A ) ) ) )
Theorem	fin45 8829	Every IV-finite set is V-finite: if we can pack two copies of the set into itself, we can certainly leave space. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 18-May-2015.)
|- ( A e. FinIV -> A e. FinV )
Theorem	fin56 8830	Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. FinV -> A e. FinVI )
Theorem	fin17 8831	Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin -> A e. FinVII )
Theorem	fin67 8832	Every VI-finite set is VII-finite. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. FinVI -> A e. FinVII )
Theorem	isfin7-2 8833	A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
|- ( A e. V -> ( A e. FinVII <-> ( A e. dom card -> A e. Fin ) ) )
Theorem	fin71num 8834	A well-orderable set is VII-finite iff it is I-finite. Thus, even without choice, on the class of well-orderable sets all eight definitions of finite set coincide. (Contributed by Mario Carneiro, 18-May-2015.)
|- ( A e. dom card -> ( A e. FinVII <-> A e. Fin ) )
Theorem	dffin7-2 8835	Class form of isfin7-2 8833. (Contributed by Mario Carneiro, 17-May-2015.)
|- FinVII = ( Fin u. ( _V \ dom card ) )
Theorem	dfacfin7 8836	Axiom of Choice equivalent: the VII-finite sets are the same as I-finite sets. (Contributed by Mario Carneiro, 18-May-2015.)
|- (CHOICE <-> FinVII = Fin )
Theorem	fin1a2lem1 8837	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- S = ( x e. On |-> suc x )   =>    |- ( A e. On -> ( S ` A ) = suc A )
Theorem	fin1a2lem2 8838	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- S = ( x e. On |-> suc x )   =>    |- S : On -1-1-> On
Theorem	fin1a2lem3 8839	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- E = ( x e. om |-> ( 2o .o x ) )   =>    |- ( A e. om -> ( E ` A ) = ( 2o .o A ) )
Theorem	fin1a2lem4 8840	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- E = ( x e. om |-> ( 2o .o x ) )   =>    |- E : om -1-1-> om
Theorem	fin1a2lem5 8841	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- E = ( x e. om |-> ( 2o .o x ) )   =>    |- ( A e. om -> ( A e. ran E <-> -. suc A e. ran E ) )
Theorem	fin1a2lem6 8842	Lemma for fin1a2 8852. Establish that om can be broken into two equipollent pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- E = ( x e. om |-> ( 2o .o x ) )   &    |- S = ( x e. On |-> suc x )   =>    |- ( S |` ran E ) : ran E -1-1-onto-> ( om \ ran E )
Theorem	fin1a2lem7 8843*	Lemma for fin1a2 8852. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- E = ( x e. om |-> ( 2o .o x ) )   &    |- S = ( x e. On |-> suc x )   =>    |- ( ( A e. V /\ A. y e. ~P A ( y e. FinIII \/ ( A \ y ) e. FinIII ) ) -> A e. FinIII )
Theorem	fin1a2lem8 8844*	Lemma for fin1a2 8852. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- ( ( A e. V /\ A. x e. ~P A ( x e. FinIII \/ ( A \ x ) e. FinIII ) ) -> A e. FinIII )
Theorem	fin1a2lem9 8845*	Lemma for fin1a2 8852. In a chain of finite sets, initial segments are finite. (Contributed by Stefan O'Rear, 8-Nov-2014.)
|- ( ( [ C.] Or X /\ X C_ Fin /\ A e. om ) -> { b e. X | b ~<_ A } e. Fin )
Theorem	fin1a2lem10 8846	Lemma for fin1a2 8852. A nonempty finite union of members of a chain is a member of the chain. (Contributed by Stefan O'Rear, 8-Nov-2014.)
|- ( ( A =/= (/) /\ A e. Fin /\ [ C.] Or A ) -> U. A e. A )
Theorem	fin1a2lem11 8847*	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 8-Nov-2014.)
|- ( ( [ C.] Or A /\ A C_ Fin ) -> ran ( b e. om |-> U. { c e. A | c ~<_ b } ) = ( A u. { (/) } ) )
Theorem	fin1a2lem12 8848	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( ( ( A C_ ~P B /\ [ C.] Or A /\ -. U. A e. A ) /\ ( A C_ Fin /\ A =/= (/) ) ) -> -. B e. FinIII )
Theorem	fin1a2lem13 8849	Lemma for fin1a2 8852. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( ( ( A C_ ~P B /\ [ C.] Or A /\ -. U. A e. A ) /\ ( -. C e. Fin /\ C e. A ) ) -> -. ( B \ C ) e. FinII )
Theorem	fin12 8850	Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8852. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin -> A e. FinII )
Theorem	fin1a2s 8851*	An II-infinite set can have an I-infinite part broken off and remain II-infinite. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- ( ( A e. V /\ A. x e. ~P A ( x e. Fin \/ ( A \ x ) e. FinII ) ) -> A e. FinII )
Theorem	fin1a2 8852	Every Ia-finite set is II-finite. Theorem 1 of [Levy58], p. 3. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
|- ( A e. FinIa -> A e. FinII )
2.6.13  Hereditarily size-limited sets without Choice
Theorem	itunifval 8853*	Function value of iterated unions. EDITORIAL: The iterated unions and order types of ordered sets are split out here because they could conceivably be independently useful. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   =>    |- ( A e. V -> ( U ` A ) = ( rec ( ( y e. _V |-> U. y ) , A ) |` om ) )
Theorem	itunifn 8854*	Functionality of the iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   =>    |- ( A e. V -> ( U ` A ) Fn om )
Theorem	ituni0 8855*	A zero-fold iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   =>    |- ( A e. V -> ( ( U ` A ) ` (/) ) = A )
Theorem	itunisuc 8856*	Successor iterated union. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   =>    |- ( ( U ` A ) ` suc B ) = U. ( ( U ` A ) ` B )
Theorem	itunitc1 8857*	Each union iterate is a member of the transitive closure. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   =>    |- ( ( U ` A ) ` B ) C_ ( TC ` A )
Theorem	itunitc 8858*	The union of all union iterates creates the transitive closure; compare trcl 8220. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   =>    |- ( TC ` A ) = U. ran ( U ` A )
Theorem	ituniiun 8859*	Unwrap an iterated union from the "other end". (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   =>    |- ( A e. V -> ( ( U ` A ) ` suc B ) = U_ a e. A ( ( U ` a ) ` B ) )
Theorem	hsmexlem7 8860*	Lemma for hsmex 8869. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
|- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )   =>    |- ( H ` (/) ) = (har ` ~P X )
Theorem	hsmexlem8 8861*	Lemma for hsmex 8869. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
|- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )   =>    |- ( a e. om -> ( H ` suc a ) = (har ` ~P ( X X. ( H ` a ) ) ) )
Theorem	hsmexlem9 8862*	Lemma for hsmex 8869. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
|- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )   =>    |- ( a e. om -> ( H ` a ) e. On )
Theorem	hsmexlem1 8863	Lemma for hsmex 8869. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- O = OrdIso ( _E , A )   =>    |- ( ( A C_ On /\ A ~<_* B ) -> dom O e. (har ` ~P B ) )
Theorem	hsmexlem2 8864*	Lemma for hsmex 8869. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9007 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- F = OrdIso ( _E , B )   &    |- G = OrdIso ( _E , U_ a e. A B )   =>    |- ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. (har ` ~P ( A X. C ) ) )
Theorem	hsmexlem3 8865*	Lemma for hsmex 8869. Clear I hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g. using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- F = OrdIso ( _E , B )   &    |- G = OrdIso ( _E , U_ a e. A B )   =>    |- ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. (har ` ~P ( D X. C ) ) )
Theorem	hsmexlem4 8866*	Lemma for hsmex 8869. The core induction, establishing bounds on the order types of iterated unions of the initial set. (Contributed by Stefan O'Rear, 14-Feb-2015.)
|- X e. _V   &    |- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )   &    |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   &    |- S = { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X }   &    |- O = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )   =>    |- ( ( c e. om /\ d e. S ) -> dom O e. ( H ` c ) )
Theorem	hsmexlem5 8867*	Lemma for hsmex 8869. Combining the above constraints, along with itunitc 8858 and tcrank 8363, gives an effective constraint on the rank of S. (Contributed by Stefan O'Rear, 14-Feb-2015.)
|- X e. _V   &    |- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )   &    |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   &    |- S = { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X }   &    |- O = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )   =>    |- ( d e. S -> ( rank ` d ) e. (har ` ~P ( om X. U. ran H ) ) )
Theorem	hsmexlem6 8868*	Lemma for hsmex 8869. (Contributed by Stefan O'Rear, 14-Feb-2015.)
|- X e. _V   &    |- H = ( rec ( ( z e. _V |-> (har ` ~P ( X X. z ) ) ) , (har ` ~P X ) ) |` om )   &    |- U = ( x e. _V |-> ( rec ( ( y e. _V |-> U. y ) , x ) |` om ) )   &    |- S = { a e. U. ( R1 " On ) | A. b e. ( TC ` { a } ) b ~<_ X }   &    |- O = OrdIso ( _E , ( rank " ( ( U ` d ) ` c ) ) )   =>    |- S e. _V
Theorem	hsmex 8869*	The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 8116. (Contributed by Stefan O'Rear, 14-Feb-2015.)
|- ( X e. V -> { s e. U. ( R1 " On ) | A. x e. ( TC ` { s } ) x ~<_ X } e. _V )
Theorem	hsmex2 8870*	The set of hereditary size-limited sets, assuming ax-reg 8116. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- ( X e. V -> { s | A. x e. ( TC ` { s } ) x ~<_ X } e. _V )
Theorem	hsmex3 8871*	The set of hereditary size-limited sets, assuming ax-reg 8116, using strict comparison (an easy corollary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- ( X e. V -> { s | A. x e. ( TC ` { s } ) x ~< X } e. _V )
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY In this section we add the Axiom of Choice ax-ac 8896, as well as weaker forms such as the axiom of countable choice ax-cc 8872 and dependent choice ax-dc 8883. We introduce these weaker forms so that theorems that do not need the full power of the axiom of choice, but need more than simple ZF, can use these intermediate axioms instead. The combination of the Zermelo-Fraenkel axioms and the axiom of choice is often abbreviated as ZFC. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. The axiom of choice does not satisfy those who wish to have a constructive proof (e.g., it will not satisfy intuitionistic logic). Thus, we make it easy to identify which proofs depend on the axiom of choice or its weaker forms.
3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
3.1.1  Introduce the Axiom of Countable Choice
Axiom	ax-cc 8872*	The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8914, but is weak enough that it can be proven using DC (see axcc 8895). It is, however, strictly stronger than ZF and cannot be proven in ZF. It states that any countable collection of nonempty sets must have a choice function. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- ( x ~~ om -> E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) )
Theorem	axcc2lem 8873*	Lemma for axcc2 8874. (Contributed by Mario Carneiro, 8-Feb-2013.)
|- K = ( n e. om |-> if ( ( F ` n ) = (/) , { (/) } , ( F ` n ) ) )   &    |- A = ( n e. om |-> ( { n } X. ( K ` n ) ) )   &    |- G = ( n e. om |-> ( 2nd ` ( f ` ( A ` n ) ) ) )   =>    |- E. g ( g Fn om /\ A. n e. om ( ( F ` n ) =/= (/) -> ( g ` n ) e. ( F ` n ) ) )
Theorem	axcc2 8874*	A possibly more useful version of ax-cc using sequences instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.)
|- E. g ( g Fn om /\ A. n e. om ( ( F ` n ) =/= (/) -> ( g ` n ) e. ( F ` n ) ) )
Theorem	axcc3 8875*	A possibly more useful version of ax-cc 8872 using sequences F ( n ) instead of countable sets. The Axiom of Infinity is needed to prove this, and indeed this implies the Axiom of Infinity. (Contributed by Mario Carneiro, 8-Feb-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)
|- F e. _V   &    |- N ~~ om   =>    |- E. f ( f Fn N /\ A. n e. N ( F =/= (/) -> ( f ` n ) e. F ) )
Theorem	axcc4 8876*	A version of axcc3 8875 that uses wffs instead of classes. (Contributed by Mario Carneiro, 7-Apr-2013.)
|- A e. _V   &    |- N ~~ om   &    |- ( x = ( f ` n ) -> ( ph <-> ps ) )   =>    |- ( A. n e. N E. x e. A ph -> E. f ( f : N --> A /\ A. n e. N ps ) )
Theorem	acncc 8877	An ax-cc 8872 equivalent: every set has choice sets of length om. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- AC om = _V
Theorem	axcc4dom 8878*	Relax the constraint on axcc4 8876 to dominance instead of equinumerosity. (Contributed by Mario Carneiro, 18-Jan-2014.)
|- A e. _V   &    |- ( x = ( f ` n ) -> ( ph <-> ps ) )   =>    |- ( ( N ~<_ om /\ A. n e. N E. x e. A ph ) -> E. f ( f : N --> A /\ A. n e. N ps ) )
Theorem	domtriomlem 8879*	Lemma for domtriom 8880. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   &    |- B = { y | ( y C_ A /\ y ~~ ~P n ) }   &    |- C = ( n e. om |-> ( ( b ` n ) \ U_ k e. n ( b ` k ) ) )   =>    |- ( -. A e. Fin -> om ~<_ A )
Theorem	domtriom 8880	Trichotomy of equinumerosity for om, proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 8751) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   =>    |- ( om ~<_ A <-> -. A ~< om )
Theorem	fin41 8881	Under countable choice, the IV-finite sets (Dedekind-finite) coincide with I-finite (finite in the usual sense) sets. (Contributed by Mario Carneiro, 16-May-2015.)
|- FinIV = Fin
Theorem	dominf 8882	A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8872. See dominfac 9005 for a version proved from ax-ac 8896. The axiom of Regularity is used for this proof, via inf3lem6 8147, and its use is necessary: otherwise the set A = { A } or A = { (/) , A } (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   =>    |- ( ( A =/= (/) /\ A C_ U. A ) -> om ~<_ A )
3.1.2  Introduce the Axiom of Dependent Choice
Axiom	ax-dc 8883*	Dependent Choice. Axiom DC1 of [Schechter] p. 149. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. Dependent choice is equivalent to the statement that every (nonempty) pruned tree has a branch. This axiom is redundant in ZFC; see axdc 8958. But ZF+DC is strictly weaker than ZF+AC, so this axiom provides for theorems that do not need the full power of AC. (Contributed by Mario Carneiro, 25-Jan-2013.)
|- ( ( E. y E. z y x z /\ ran x C_ dom x ) -> E. f A. n e. om ( f ` n ) x ( f ` suc n ) )
Theorem	dcomex 8884	The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
|- om e. _V
Theorem	axdc2lem 8885*	Lemma for axdc2 8886. We construct a relation R based on F such that x R y iff y e. ( F ` x ), and show that the "function" described by ax-dc 8883 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). (Contributed by Mario Carneiro, 25-Jan-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- A e. _V   &    |- R = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }   &    |- G = ( x e. om |-> ( h ` x ) )   =>    |- ( ( A =/= (/) /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )
Theorem	axdc2 8886*	An apparent strengthening of ax-dc 8883 (but derived from it) which shows that there is a denumerable sequence g for any function that maps elements of a set A to nonempty subsets of A such that g ( x + 1 ) e. F ( g ( x ) ) for all x e. om. The finitistic version of this can be proven by induction, but the infinite version requires this new axiom. (Contributed by Mario Carneiro, 25-Jan-2013.)
|- A e. _V   =>    |- ( ( A =/= (/) /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )
Theorem	axdc3lem 8887*	The class S of finite approximations to the DC sequence is a set. (We derive here the stronger statement that S is a subset of a specific set, namely ~P ( om X. A ).) (Unnecessary distinct variable restrictions were removed by David Abernethy, 18-Mar-2014.) (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario Carneiro, 18-Mar-2014.)
|- A e. _V   &    |- S = { s | E. n e. om ( s : suc n --> A /\ ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) }   =>    |- S e. _V
Theorem	axdc3lem2 8888*	Lemma for axdc3 8891. We have constructed a "candidate set" S, which consists of all finite sequences s that satisfy our property of interest, namely s ( x + 1 ) e. F ( s ( x ) ) on its domain, but with the added constraint that s ( 0 ) = C. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 8883 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely ( h ` n ) : m --> A (for some integer m). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 8883 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence h, we can construct the sequence g that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
|- A e. _V   &    |- S = { s | E. n e. om ( s : suc n --> A /\ ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) }   &    |- G = ( x e. S |-> { y e. S | ( dom y = suc dom x /\ ( y |` dom x ) = x ) } )   =>    |- ( E. h ( h : om --> S /\ A. k e. om ( h ` suc k ) e. ( G ` ( h ` k ) ) ) -> E. g ( g : om --> A /\ ( g ` (/) ) = C /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )
Theorem	axdc3lem3 8889*	Simple substitution lemma for axdc3 8891. (Contributed by Mario Carneiro, 27-Jan-2013.)
|- A e. _V   &    |- S = { s | E. n e. om ( s : suc n --> A /\ ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) }   &    |- B e. _V   =>    |- ( B e. S <-> E. m e. om ( B : suc m --> A /\ ( B ` (/) ) = C /\ A. k e. m ( B ` suc k ) e. ( F ` ( B ` k ) ) ) )
Theorem	axdc3lem4 8890*	Lemma for axdc3 8891. We have constructed a "candidate set" S, which consists of all finite sequences s that satisfy our property of interest, namely s ( x + 1 ) e. F ( s ( x ) ) on its domain, but with the added constraint that s ( 0 ) = C. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 8883 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely ( h ` n ) : m --> A (for some integer m). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 8883 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that S is nonempty, and that G always maps to a nonempty subset of S, so that we can apply axdc2 8886. See axdc3lem2 8888 for the rest of the proof. (Contributed by Mario Carneiro, 27-Jan-2013.)
|- A e. _V   &    |- S = { s | E. n e. om ( s : suc n --> A /\ ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) }   &    |- G = ( x e. S |-> { y e. S | ( dom y = suc dom x /\ ( y |` dom x ) = x ) } )   =>    |- ( ( C e. A /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ ( g ` (/) ) = C /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )
Theorem	axdc3 8891*	Dependent Choice. Axiom DC1 of [Schechter] p. 149, with the addition of an initial value C. This theorem is weaker than the Axiom of Choice but is stronger than Countable Choice. It shows the existence of a sequence whose values can only be shown to exist (but cannot be constructed explicitly) and also depend on earlier values in the sequence. (Contributed by Mario Carneiro, 27-Jan-2013.)
|- A e. _V   =>    |- ( ( C e. A /\ F : A --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ ( g ` (/) ) = C /\ A. k e. om ( g ` suc k ) e. ( F ` ( g ` k ) ) ) )
Theorem	axdc4lem 8892*	Lemma for axdc4 8893. (Contributed by Mario Carneiro, 31-Jan-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- A e. _V   &    |- G = ( n e. om , x e. A |-> ( { suc n } X. ( n F x ) ) )   =>    |- ( ( C e. A /\ F : ( om X. A ) --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ ( g ` (/) ) = C /\ A. k e. om ( g ` suc k ) e. ( k F ( g ` k ) ) ) )
Theorem	axdc4 8893*	A more general version of axdc3 8891 that allows the function F to vary with k. (Contributed by Mario Carneiro, 31-Jan-2013.)
|- A e. _V   =>    |- ( ( C e. A /\ F : ( om X. A ) --> ( ~P A \ { (/) } ) ) -> E. g ( g : om --> A /\ ( g ` (/) ) = C /\ A. k e. om ( g ` suc k ) e. ( k F ( g ` k ) ) ) )
Theorem	axcclem 8894*	Lemma for axcc 8895. (Contributed by Mario Carneiro, 2-Feb-2013.) (Revised by Mario Carneiro, 16-Nov-2013.)
|- A = ( x \ { (/) } )   &    |- F = ( n e. om , y e. U. A |-> ( f ` n ) )   &    |- G = ( w e. A |-> ( h ` suc ( `' f ` w ) ) )   =>    |- ( x ~~ om -> E. g A. z e. x ( z =/= (/) -> ( g ` z ) e. z ) )
Theorem	axcc 8895*	Although CC can be proven trivially using ac5 8914, we prove it here using DC. (New usage is discouraged.) (Contributed by Mario Carneiro, 2-Feb-2013.)
|- ( x ~~ om -> E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) )
3.2  ZFC Set Theory - add the Axiom of Choice
3.2.1  Introduce the Axiom of Choice
Axiom	ax-ac 8896*	Axiom of Choice. The Axiom of Choice (AC) is usually considered an extension of ZF set theory rather than a proper part of it. It is sometimes considered philosophically controversial because it asserts the existence of a set without telling us what the set is. ZF set theory that includes AC is called ZFC. The unpublished version given here says that given any set x, there exists a y that is a collection of unordered pairs, one pair for each nonempty member of x. One entry in the pair is the member of x, and the other entry is some arbitrary member of that member of x. See the rewritten version ac3 8899 for a more detailed explanation. Theorem ac2 8898 shows an equivalent written compactly with restricted quantifiers. This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 8902 is slightly shorter when the biconditional of ax-ac 8896 is expanded into implication and negation. In axac3 8901 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9113 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 8929, ac5 8914, and ac7 8910. The Axiom of Regularity ax-reg 8116 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8568. Equivalents to AC are the well-ordering theorem weth 8932 and Zorn's lemma zorn 8944. See ac4 8912 for comments about stronger versions of AC. In order to avoid uses of ax-reg 8116 for derivation of AC equivalents, we provide ax-ac2 8900 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8900 from ax-ac 8896 is shown by theorem axac2 8903, and the reverse derivation by axac 8904. Therefore, new proofs should normally use ax-ac2 8900 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
|- E. y A. z A. w ( ( z e. w /\ w e. x ) -> E. v A. u ( E. t ( ( u e. w /\ w e. t ) /\ ( u e. t /\ t e. y ) ) <-> u = v ) )
Theorem	zfac 8897*	Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8896. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)
|- E. x A. y A. z ( ( y e. z /\ z e. w ) -> E. w A. y ( E. w ( ( y e. z /\ z e. w ) /\ ( y e. w /\ w e. x ) ) <-> y = w ) )
Theorem	ac2 8898*	Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 8899 is easier to understand.) Note: aceq0 8556 shows the logical equivalence to ax-ac 8896. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
|- E. y A. z e. x A. w e. z E! v e. z E. u e. y ( z e. u /\ v e. u )
Theorem	ac3 8899*	Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8896 can be established by chaining aceq0 8556 and aceq2 8557. A standard textbook version of AC is derived from this one in dfac2a 8567, and this version of AC is derived from the textbook version in dfac2 8568. The following sketch will help you understand this version of the axiom. Given any set x, the axiom says that there exists a y that is a collection of unordered pairs, one pair for each nonempty member of x. One entry in the pair is the member of x, and the other entry is some arbitrary member of that member of x. Using the Axiom of Regularity, we can show that y is really a set of ordered pairs, very similar to the ordered pair construction opthreg 8132. The key theorem for this (used in the proof of dfac2 8568) is preleq 8131. With this modified definition of ordered pair, it can be seen that y is actually a choice function on the members of x. For example, suppose x = { { 1 , 2 } , { 1 , 3 } , { 2 , 3 , 4 } }. Let us try y = { { { 1 , 2 } , 1 } , { { 1 , 3 } , 1 } , { { 2 , 3 , 4 } , 2 } }. For the member (of x) z = { 1 , 2 }, the only assignment to w and v that satisfies the axiom is w = 1 and v = { { 1 , 2 } , 1 }, so there is exactly one w as required. We verify the other two members of x similarly. Thus, y satisfies the axiom. Using our modified ordered pair definition, we can say that y corresponds to the choice function { <. { 1 , 2 } , 1 >. , <. { 1 , 3 } , 1 >. , <. { 2 , 3 , 4 } , 2 >. }. Of course other choices for y will also satisfy the axiom, for example y = { { { 1 , 2 } , 2 } , { { 1 , 3 } , 1 } , { { 2 , 3 , 4 } , 4 } }. What AC tells us is that there exists at least one such y, but it doesn't tell us which one. (New usage is discouraged.) (Contributed by NM, 19-Jul-1996.)
|- E. y A. z e. x ( z =/= (/) -> E! w e. z E. v e. y ( z e. v /\ w e. v ) )
Axiom	ax-ac2 8900*	In order to avoid uses of ax-reg 8116 for derivation of AC equivalents, we provide ax-ac2 8900, which is equivalent to the standard AC of textbooks. This appears to be the shortest known equivalent to the standard AC when expressed in terms of set theory primitives. It was found by Kurt Maes as theorem ackm 8902. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1663 available. The derivation of ax-ac2 8900 from ax-ac 8896 is shown by theorem axac2 8903, and the reverse derivation by axac 8904. Note that we use ax-reg 8116 to derive ax-ac 8896 from ax-ac2 8900, but not to derive ax-ac2 8900 from ax-ac 8896. (Contributed by NM, 19-Dec-2016.)
|- E. y A. z E. v A. u ( ( y e. x /\ ( z e. y -> ( ( v e. x /\ -. y = v ) /\ z e. v ) ) ) \/ ( -. y e. x /\ ( z e. x -> ( ( v e. z /\ v e. y ) /\ ( ( u e. z /\ u e. y ) -> u = v ) ) ) ) )