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	isf34lem2 8801*	Lemma for isfin3-4 8810. (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 8802*	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 8803*	Lemma for isfin3-4 8810. (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 8804*	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 8805*	Lemma for isfin3-4 8810. (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 8806*	Lemma for isfin3-4 8810. (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 8807*	Lemma for isfin3-4 8810. (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 8808*	Lemma for isfin3-4 8810. (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 8809*	Inference from isfin3-4 8810. (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 8810*	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 8811	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 8812	Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)
|- ( A ~~ B -> ( A e. FinIa -> B e. FinIa ) )
Theorem	isfin1-2 8813	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 8814	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 8815	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 8816	Compact quantifier-free version of the standard definition df-fin 7581. (Contributed by Stefan O'Rear, 6-Jan-2015.)
|- Fin = ( ~~ " om )
Theorem	fin23 8817	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 8746) 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 8874 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 8818	Every III-finite set is IV-finite. (Contributed by Stefan O'Rear, 30-Oct-2014.)
|- ( A e. FinIII -> A e. FinIV )
Theorem	isfin5-2 8819	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 8820	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 8821	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 8822	Every I-finite set is VII-finite. (Contributed by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin -> A e. FinVII )
Theorem	fin67 8823	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 8824	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 8825	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 8826	Class form of isfin7-2 8824. (Contributed by Mario Carneiro, 17-May-2015.)
|- FinVII = ( Fin u. ( _V \ dom card ) )
Theorem	dfacfin7 8827	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 8828	Lemma for fin1a2 8843. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- S = ( x e. On |-> suc x )   =>    |- ( A e. On -> ( S ` A ) = suc A )
Theorem	fin1a2lem2 8829	Lemma for fin1a2 8843. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- S = ( x e. On |-> suc x )   =>    |- S : On -1-1-> On
Theorem	fin1a2lem3 8830	Lemma for fin1a2 8843. (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 8831	Lemma for fin1a2 8843. (Contributed by Stefan O'Rear, 7-Nov-2014.)
|- E = ( x e. om |-> ( 2o .o x ) )   =>    |- E : om -1-1-> om
Theorem	fin1a2lem5 8832	Lemma for fin1a2 8843. (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 8833	Lemma for fin1a2 8843. 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 8834*	Lemma for fin1a2 8843. 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 8835*	Lemma for fin1a2 8843. 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 8836*	Lemma for fin1a2 8843. 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 8837	Lemma for fin1a2 8843. 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 8838*	Lemma for fin1a2 8843. (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 8839	Lemma for fin1a2 8843. (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 8840	Lemma for fin1a2 8843. (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 8841	Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 8843. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
|- ( A e. Fin -> A e. FinII )
Theorem	fin1a2s 8842*	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 8843	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 8844*	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 8845*	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 8846*	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 8847*	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 8848*	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 8849*	The union of all union iterates creates the transitive closure; compare trcl 8211. (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 8850*	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 8851*	Lemma for hsmex 8860. 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 8852*	Lemma for hsmex 8860. 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 8853*	Lemma for hsmex 8860. 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 8854	Lemma for hsmex 8860. 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 8855*	Lemma for hsmex 8860. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 8998 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 8856*	Lemma for hsmex 8860. 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 8857*	Lemma for hsmex 8860. 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 8858*	Lemma for hsmex 8860. Combining the above constraints, along with itunitc 8849 and tcrank 8354, 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 8859*	Lemma for hsmex 8860. (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 8860*	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 8107. (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 8861*	The set of hereditary size-limited sets, assuming ax-reg 8107. (Contributed by Stefan O'Rear, 11-Feb-2015.)
|- ( X e. V -> { s | A. x e. ( TC ` { s } ) x ~<_ X } e. _V )
Theorem	hsmex3 8862*	The set of hereditary size-limited sets, assuming ax-reg 8107, 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 8887, as well as weaker forms such as the axiom of countable choice ax-cc 8863 and dependent choice ax-dc 8874. 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 8863*	The axiom of countable choice (CC), also known as the axiom of denumerable choice. It is clearly a special case of ac5 8905, but is weak enough that it can be proven using DC (see axcc 8886). 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 8864*	Lemma for axcc2 8865. (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 8865*	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 8866*	A possibly more useful version of ax-cc 8863 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 8867*	A version of axcc3 8866 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 8868	An ax-cc 8863 equivalent: every set has choice sets of length om. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- AC om = _V
Theorem	axcc4dom 8869*	Relax the constraint on axcc4 8867 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 8870*	Lemma for domtriom 8871. (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 8871	Trichotomy of equinumerosity for om, proven using CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 8742) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.)
|- A e. _V   =>    |- ( om ~<_ A <-> -. A ~< om )
Theorem	fin41 8872	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 8873	A nonempty set that is a subset of its union is infinite. This version is proved from ax-cc 8863. See dominfac 8996 for a version proved from ax-ac 8887. The axiom of Regularity is used for this proof, via inf3lem6 8138, 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 8874*	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 8949. 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 8875	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 8876*	Lemma for axdc2 8877. 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 8874 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 8877*	An apparent strengthening of ax-dc 8874 (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 8878*	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 8879*	Lemma for axdc3 8882. 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 8874 (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 8874 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 8880*	Simple substitution lemma for axdc3 8882. (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 8881*	Lemma for axdc3 8882. 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 8874 (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 8874 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 8877. See axdc3lem2 8879 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 8882*	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 8883*	Lemma for axdc4 8884. (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 8884*	A more general version of axdc3 8882 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 8885*	Lemma for axcc 8886. (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 8886*	Although CC can be proven trivially using ac5 8905, 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 8887*	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 8890 for a more detailed explanation. Theorem ac2 8889 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 8893 is slightly shorter when the biconditional of ax-ac 8887 is expanded into implication and negation. In axac3 8892 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9105 (the Generalized Continuum Hypothesis implies the Axiom of Choice). Standard textbook versions of AC are derived as ac8 8920, ac5 8905, and ac7 8901. The Axiom of Regularity ax-reg 8107 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8559. Equivalents to AC are the well-ordering theorem weth 8923 and Zorn's lemma zorn 8935. See ac4 8903 for comments about stronger versions of AC. In order to avoid uses of ax-reg 8107 for derivation of AC equivalents, we provide ax-ac2 8891 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8891 from ax-ac 8887 is shown by theorem axac2 8894, and the reverse derivation by axac 8895. Therefore, new proofs should normally use ax-ac2 8891 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 8888*	Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8887. (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 8889*	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 8890 is easier to understand.) Note: aceq0 8547 shows the logical equivalence to ax-ac 8887. (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 8890*	Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8887 can be established by chaining aceq0 8547 and aceq2 8548. A standard textbook version of AC is derived from this one in dfac2a 8558, and this version of AC is derived from the textbook version in dfac2 8559. 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 8123. The key theorem for this (used in the proof of dfac2 8559) is preleq 8122. 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 8891*	In order to avoid uses of ax-reg 8107 for derivation of AC equivalents, we provide ax-ac2 8891, 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 8893. We removed the leading quantifier to make it slightly shorter, since we have ax-gen 1665 available. The derivation of ax-ac2 8891 from ax-ac 8887 is shown by theorem axac2 8894, and the reverse derivation by axac 8895. Note that we use ax-reg 8107 to derive ax-ac 8887 from ax-ac2 8891, but not to derive ax-ac2 8891 from ax-ac 8887. (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 ) ) ) ) )
Theorem	axac3 8892	This theorem asserts that the constant CHOICE is a theorem, thus eliminating it as a hypothesis while assuming ax-ac2 8891 as an axiom. (Contributed by Mario Carneiro, 6-May-2015.) (Revised by NM, 20-Dec-2016.) (Proof modification is discouraged.)
|- CHOICE
Theorem	ackm 8893*	A remarkable equivalent to the Axiom of Choice that has only five quantifiers (when expanded to e., = primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem dfackm 8594. Maes found this version of AC in April, 2004 (replacing a longer version, also with five quantifiers, that he found in November, 2003). See Kurt Maes, "A 5-quantifier ( e.,=)-expression ZF-equivalent to the Axiom of Choice" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf). The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)
|- A. x 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 ) ) ) ) )
Theorem	axac2 8894*	Derive ax-ac2 8891 from ax-ac 8887. (Contributed by NM, 19-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
|- 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 ) ) ) ) )
Theorem	axac 8895*	Derive ax-ac 8887 from ax-ac2 8891. Note that ax-reg 8107 is used by the proof. (Contributed by NM, 19-Dec-2016.) (Proof modification is discouraged.)
|- 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	axaci 8896	Apply a choice equivalent. (Contributed by Mario Carneiro, 17-May-2015.)
|- (CHOICE <-> A. x ph )   =>    |- ph
Theorem	cardeqv 8897	All sets are well-orderable under choice. (Contributed by Mario Carneiro, 28-Apr-2015.)
|- dom card = _V
Theorem	numth3 8898	All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.)
|- ( A e. V -> A e. dom card )
Theorem	numth2 8899*	Numeration theorem: any set is equinumerous to some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 20-Oct-2003.)
|- A e. _V   =>    |- E. x e. On x ~~ A
Theorem	numth 8900*	Numeration theorem: every set can be put into one-to-one correspondence with some ordinal (using AC). Theorem 10.3 of [TakeutiZaring] p. 84. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
|- A e. _V   =>    |- E. x e. On E. f f : x -1-1-onto-> A