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	isf32lem1 8801*	Lemma for isfin3-2 8815. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   =>    |- ( ( ( A e. om /\ B e. om ) /\ ( B C_ A /\ ph ) ) -> ( F ` A ) C_ ( F ` B ) )
Theorem	isf32lem2 8802*	Lemma for isfin3-2 8815. Non-minimum implies that there is always another decrease. (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   =>    |- ( ( ph /\ A e. om ) -> E. a e. om ( A e. a /\ ( F ` suc a ) C. ( F ` a ) ) )
Theorem	isf32lem3 8803*	Lemma for isfin3-2 8815. Being a chain, difference sets are disjoint (one case). (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   =>    |- ( ( ( A e. om /\ B e. om ) /\ ( B e. A /\ ph ) ) -> ( ( ( F ` A ) \ ( F ` suc A ) ) i^i ( ( F ` B ) \ ( F ` suc B ) ) ) = (/) )
Theorem	isf32lem4 8804*	Lemma for isfin3-2 8815. Being a chain, difference sets are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   =>    |- ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) -> ( ( ( F ` A ) \ ( F ` suc A ) ) i^i ( ( F ` B ) \ ( F ` suc B ) ) ) = (/) )
Theorem	isf32lem5 8805*	Lemma for isfin3-2 8815. There are infinite decrease points. (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   &    |- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }   =>    |- ( ph -> -. S e. Fin )
Theorem	isf32lem6 8806*	Lemma for isfin3-2 8815. Each K value is nonempty. (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   &    |- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }   &    |- J = ( u e. om |-> ( iota_ v e. S ( v i^i S ) ~~ u ) )   &    |- K = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J )   =>    |- ( ( ph /\ A e. om ) -> ( K ` A ) =/= (/) )
Theorem	isf32lem7 8807*	Lemma for isfin3-2 8815. Different K values are disjoint. (Contributed by Stefan O'Rear, 5-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   &    |- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }   &    |- J = ( u e. om |-> ( iota_ v e. S ( v i^i S ) ~~ u ) )   &    |- K = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J )   =>    |- ( ( ( ph /\ A =/= B ) /\ ( A e. om /\ B e. om ) ) -> ( ( K ` A ) i^i ( K ` B ) ) = (/) )
Theorem	isf32lem8 8808*	Lemma for isfin3-2 8815. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   &    |- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }   &    |- J = ( u e. om |-> ( iota_ v e. S ( v i^i S ) ~~ u ) )   &    |- K = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J )   =>    |- ( ( ph /\ A e. om ) -> ( K ` A ) C_ G )
Theorem	isf32lem9 8809*	Lemma for isfin3-2 8815. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   &    |- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }   &    |- J = ( u e. om |-> ( iota_ v e. S ( v i^i S ) ~~ u ) )   &    |- K = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J )   &    |- L = ( t e. G |-> ( iota s ( s e. om /\ t e. ( K ` s ) ) ) )   =>    |- ( ph -> L : G -onto-> om )
Theorem	isf32lem10 8810*	Lemma for isfin3-2 . Write in terms of weak dominance. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( ph -> F : om --> ~P G )   &    |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )   &    |- ( ph -> -. |^| ran F e. ran F )   &    |- S = { y e. om | ( F ` suc y ) C. ( F ` y ) }   &    |- J = ( u e. om |-> ( iota_ v e. S ( v i^i S ) ~~ u ) )   &    |- K = ( ( w e. S |-> ( ( F ` w ) \ ( F ` suc w ) ) ) o. J )   &    |- L = ( t e. G |-> ( iota s ( s e. om /\ t e. ( K ` s ) ) ) )   =>    |- ( ph -> ( G e. V -> om ~<_* G ) )
Theorem	isf32lem11 8811*	Lemma for isfin3-2 8815. Remove hypotheses from isf32lem10 8810. (Contributed by Stefan O'Rear, 17-May-2015.)
|- ( ( G e. V /\ ( F : om --> ~P G /\ A. b e. om ( F ` suc b ) C_ ( F ` b ) /\ -. |^| ran F e. ran F ) ) -> om ~<_* G )
Theorem	isf32lem12 8812*	Lemma for isfin3-2 8815. (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 8813	One half of isfin3-2 8815. (Contributed by Mario Carneiro, 3-Jun-2015.)
|- ( A e. FinIII -> -. om ~<_* A )
Theorem	isf33lem 8814*	Lemma for isfin3-3 8816. (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 8815	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 8816*	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 8817*	Inference from isfin3-3 8816. (This is actually a bit stronger than isfin3-3 8816 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 8818*	Lemma for compsscnv 8819. (Contributed by Mario Carneiro, 17-May-2015.)
|- ( ( x e. ~P A /\ y = ( A \ x ) ) -> ( y e. ~P A /\ x = ( A \ y ) ) )
Theorem	compsscnv 8819*	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 8820*	Lemma for isfin3-4 8830. (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 8821*	Lemma for isfin3-4 8830. (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 8822*	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 8823*	Lemma for isfin3-4 8830. (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 8824*	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 8825*	Lemma for isfin3-4 8830. (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 8826*	Lemma for isfin3-4 8830. (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 8827*	Lemma for isfin3-4 8830. (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 8828*	Lemma for isfin3-4 8830. (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 8829*	Inference from isfin3-4 8830. (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 8830*	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 8831	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 8832	Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
|- ( A ~~ B -> ( A e. FinIa -> B e. FinIa ) )
Theorem	isfin1-2 8833	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 8834	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 8835	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 8836	Compact quantifier-free version of the standard definition df-fin 7591. (Contributed by Stefan O'Rear, 6-Jan-2015.)
|- Fin = ( ~~ " om )
Theorem	fin23 8837	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 8766) 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 8894 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 8838	Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
|- ( A e. FinIII -> A e. FinIV )
Theorem	isfin5-2 8839	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 8840	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 8841	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 8842	Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin -> A e. FinVII )
Theorem	fin67 8843	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 8844	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 8845	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 8846	Class form of isfin7-2 8844. (Contributed by Mario Carneiro, 17-May-2015.)
|- FinVII = ( Fin u. ( _V \ dom card ) )
Theorem	dfacfin7 8847	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 8848	Lemma for fin1a2 8863. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- S = ( x e. On |-> suc x )   =>    |- ( A e. On -> ( S ` A ) = suc A )
Theorem	fin1a2lem2 8849	Lemma for fin1a2 8863. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- S = ( x e. On |-> suc x )   =>    |- S : On -1-1-> On
Theorem	fin1a2lem3 8850	Lemma for fin1a2 8863. (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 8851	Lemma for fin1a2 8863. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- E = ( x e. om |-> ( 2o .o x ) )   =>    |- E : om -1-1-> om
Theorem	fin1a2lem5 8852	Lemma for fin1a2 8863. (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 8853	Lemma for fin1a2 8863. 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 8854*	Lemma for fin1a2 8863. 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 8855*	Lemma for fin1a2 8863. 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 8856*	Lemma for fin1a2 8863. 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 8857	Lemma for fin1a2 8863. 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 8858*	Lemma for fin1a2 8863. (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 8859	Lemma for fin1a2 8863. (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 8860	Lemma for fin1a2 8863. (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 8861	Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8863. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin -> A e. FinII )
Theorem	fin1a2s 8862*	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 8863	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 8864*	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 8865*	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 8866*	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 8867*	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 8868*	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 8869*	The union of all union iterates creates the transitive closure; compare trcl 8230. (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 8870*	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 8871*	Lemma for hsmex 8880. 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 8872*	Lemma for hsmex 8880. 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 8873*	Lemma for hsmex 8880. 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 8874	Lemma for hsmex 8880. 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 8875*	Lemma for hsmex 8880. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9018 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 8876*	Lemma for hsmex 8880. 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 8877*	Lemma for hsmex 8880. 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 8878*	Lemma for hsmex 8880. Combining the above constraints, along with itunitc 8869 and tcrank 8373, 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 8879*	Lemma for hsmex 8880. (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 8880*	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 8125. (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 8881*	The set of hereditary size-limited sets, assuming ax-reg 8125. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- ( X e. V -> { s | A. x e. ( TC ` { s } ) x ~<_ X } e. _V )
Theorem	hsmex3 8882*	The set of hereditary size-limited sets, assuming ax-reg 8125, 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 8907, as well as weaker forms such as the axiom of countable choice ax-cc 8883 and dependent choice ax-dc 8894. 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 8883*	The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8925, but is weak enough that it can be proven using DC (see axcc 8906). 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 8884*	Lemma for axcc2 8885. (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 8885*	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 8886*	A possibly more useful version of ax-cc 8883 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 8887*	A version of axcc3 8886 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 8888	An ax-cc 8883 equivalent: every set has choice sets of length om. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- AC om = _V
Theorem	axcc4dom 8889*	Relax the constraint on axcc4 8887 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 8890*	Lemma for domtriom 8891. (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 8891	Trichotomy of equinumerosity for om, proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 8762) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   =>    |- ( om ~<_ A <-> -. A ~< om )
Theorem	fin41 8892	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 8893	A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8883. See dominfac 9016 for a version proved from ax-ac 8907. The axiom of Regularity is used for this proof, via inf3lem6 8156, 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 8894*	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 8969. 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 8895	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 8896*	Lemma for axdc2 8897. 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 8894 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 8897*	An apparent strengthening of ax-dc 8894 (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 8898*	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 8899*	Lemma for axdc3 8902. 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 8894 (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 8894 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 8900*	Simple substitution lemma for axdc3 8902. (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 ) ) ) )