mmtheorems54 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5301-5400 - Page 54 of 195

Type	Label	Description
Statement
Theorem	smoeq 5301	Equality theorem for strictly monotone functions. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- (A = B -> (Smo A <-> Smo B))
Theorem	smodm 5302	The domain of a strictly monotone function is an ordinal. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- (Smo A -> Ord dom A)
Theorem	smores 5303	A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- ((Smo A /\ B e. dom A) -> Smo (A |` B))
Theorem	smores3 5304	A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)
|- ((Smo (A |` B) /\ C e. (dom A i^i B) /\ Ord B) -> Smo (A |` C))
Theorem	smores2 5305	A strictly monotone ordinal function restricted to an ordinal is still monotone.
|- ((Smo F /\ Ord A) -> Smo (F |` A))
Theorem	smodm2 5306	The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- ((F Fn A /\ Smo F) -> Ord A)
Theorem	smofvon2 5307	The function values of a strictly monotone ordinal function are ordinals. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- (Smo F -> (F` B) e. On)
Theorem	iordsmo 5308	The identity relation restricted to the ordinals is a strictly monotone function. (Contributed by Andrew Salmon, 16-Nov-2011.)
|- Ord A   =>   |- Smo ( _I |` A)
Theorem	smo0 5309	The null set is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 20-Nov-2011.)
|- Smo (/)
Theorem	smofvon 5310	If B is a strictly monotone ordinal function, and A is in the domain of B, then the value of the function at A is an ordinal. (Contributed by Andrew Salmon, 20-Nov-2011.)
|- ((Smo B /\ A e. dom B) -> (B` A) e. On)
Theorem	smoel 5311	If x is less than y then a strictly monotone function's value will be strictly less at x than at y. (Contributed by Andrew Salmon, 22-Nov-2011.)
|- ((Smo B /\ A e. dom B /\ C e. A) -> (B` C) e. (B` A))
Theorem	smoiun 5312	The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.)
|- ((Smo B /\ A e. dom B) -> U_x e. A (B` x) C_ (B` A))
Theorem	smoiso 5313	If F is an isomorphism from an ordinal A onto B, which is a subset of the ordinals, then F is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)
|- ((F Isom _E , _E (A, B) /\ Ord A /\ B C_ On) -> Smo F)
Theorem	smoel2 5314	A strictly monotone ordinal function preserves the epsilon relation. (Contributed by Mario Carneiro, 12-Mar-2013.)
|- (((F Fn A /\ Smo F) /\ (B e. A /\ C e. B)) -> (F` C) e. (F` B))
Theorem	smo11 5315	A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ((F:A-->B /\ Smo F) -> F:A-1-1->B)
Theorem	smoord 5316	A strictly monotone ordinal function preserves strict ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
|- (((F Fn A /\ Smo F) /\ (C e. A /\ D e. A)) -> (C e. D <-> (F` C) e. (F` D)))
Theorem	smoword 5317	A strictly monotone ordinal function preserves weak ordering. (Contributed by Mario Carneiro, 4-Mar-2013.)
|- (((F Fn A /\ Smo F) /\ (C e. A /\ D e. A)) -> (C C_ D <-> (F` C) C_ (F` D)))
Theorem	smogt 5318	A strictly monotone ordinal function is greater than or equal to its argument. (Contributed by Mario Carneiro, 28-Feb-2013.)
|- ((F Fn A /\ Smo F /\ C e. A) -> C C_ (F` C))
Theorem	smorndom 5319	The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013.)
|- ((F:A-->B /\ Smo F /\ Ord B) -> A C_ B)
Theorem	smoiso2 5320	The strictly monotone ordinal functions are also epsilon isomorphisms of subclasses of On.
|- ((Ord A /\ B C_ On) -> ((F:A-onto->B /\ Smo F) <-> F Isom _E , _E (A, B)))
Theorem	smoge 5321	A strictly monotonic function is always increasing. Exercise 1 in [TakeutiZaring] p. 50. It holds even when C is transfinite. (Contributed by Andrew Salmon, 23-Nov-2011.)
|- dom A = B   &   |- Smo A   =>   |- (C e. B -> C C_ (A` C))
Transfinite recursion
Theorem	tfrlem1 5322	A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|- (A e. On -> ((F Fn A /\ G Fn A) -> (A.x e. A ((F` x) = (B` (F |` x)) /\ (G` x) = (B` (G |` x))) -> A.x e. A (F` x) = (G` x))))
Theorem	tfrlem2 5323	Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 5322 into the main proof. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) [Auxiliary lemma - not displayed.]
Theorem	tfrlem3 5324	Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. [Auxiliary lemma - not displayed.]
Theorem	tfrlem3a 5325	Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. [Auxiliary lemma - not displayed.]
Theorem	tfrlem4 5326	Lemma for transfinite recursion. A is the class of all "acceptable" functions, and F is their union. First we show that an acceptable function is in fact a function. [Auxiliary lemma - not displayed.]
Theorem	tfrlem5 5327	Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. [Auxiliary lemma - not displayed.]
Theorem	tfrlem6 5328	Lemma for transfinite recursion. The union of all acceptable functions is a relation. [Auxiliary lemma - not displayed.]
Theorem	tfrlem7 5329	Lemma for transfinite recursion. The union of all acceptable functions is a function. [Auxiliary lemma - not displayed.]
Theorem	tfrlem8 5330	Lemma for transfinite recursion. The domain of F is ordinal. (The proof was shortened by Alan Sare, 11-Mar-2008.) [Auxiliary lemma - not displayed.]
Theorem	tfrlem9 5331	Lemma for transfinite recursion. Here we compute the value of F (the union of all acceptable functions). [Auxiliary lemma - not displayed.]
Theorem	tfrlem10 5332	Lemma for transfinite recursion. We define class C by extending F with one ordered pair. We will assume, falsely, that domain of F is a member of, and thus not equal to, On. Using this assumption we will prove facts about C that will lead to a contradiction in tfrlem13 5335, thus showing the domain of F does in fact equal On. Here we show (under the false assumption) that C is a function extending the domain of F by one. (The proof was shortened by Alan Sare, 20-Feb-2008.) [Auxiliary lemma - not displayed.]
Theorem	tfrlem11 5333	Lemma for transfinite recursion. Compute the value of C. [Auxiliary lemma - not displayed.]
Theorem	tfrlem12 5334	Lemma for transfinite recursion. Show C is an acceptable function. [Auxiliary lemma - not displayed.]
Theorem	tfrlem13 5335	Lemma for transfinite recursion. If dom F is in On, then C is acceptable, and thus a subset of F, but dom C is bigger than dom F. This is a contradiction, so dom F must be On. [Auxiliary lemma - not displayed.]
Theorem	tfr1 5336	Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class G, normally a function, and define a class A of all "acceptable" functions. The final function we're interested in is the union F of them. F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of F. In this first part we show that F is a function whose domain is all ordinal numbers.
|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}   &   |- F = U.A   =>   |- F Fn On
Theorem	tfr2 5337	Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function F has the property that for any function G whatsoever, the "next" value of F is G recursively applied to all "previous" values of F.
|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}   &   |- F = U.A   =>   |- (z e. On -> (F` z) = (G` (F |` z)))
Theorem	tfr3 5338	Principle of Transfinite Recursion, part 3 of 3. Theorem 7.41(3) of [TakeutiZaring] p. 47. Finally we show that F is unique. We do this by showing that any class B with the same properties of F that we showed in parts 1 and 2 is identical to F.
|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}   &   |- F = U.A   =>   |- ((B Fn On /\ A.x e. On (B` x) = (G` (B |` x))) -> B = F)
Theorem	tz7.44lem1 5339	G is a function. Lemma for tz7.44-1 5340, tz7.44-2 5341, and tz7.44-3 5342. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   =>   |- Fun G
Theorem	tz7.44-1 5340	The value of F at (/). Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49.
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   &   |- F Fn On   &   |- (x e. On -> (F` x) = (G` (F |` x)))   &   |- A e. _V   =>   |- (F` (/)) = A
Theorem	tz7.44-2 5341	The value of F at a successor ordinal. Part 2 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   &   |- F Fn On   &   |- (x e. On -> (F` x) = (G` (F |` x)))   &   |- B e. On   =>   |- (F` suc B) = (H` (F` B))
Theorem	tz7.44-3 5342	The value of F at a limit ordinal. Part 3 of Theorem 7.44 of [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|- G = {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))}   &   |- F Fn On   &   |- (x e. On -> (F` x) = (G` (F |` x)))   &   |- B e. On   =>   |- (Lim B -> (F` B) = U.(F"B))
Recursive definition generator
Syntax	crdg 5343	Extend class notation with the recursive definition generator.
class rec(A, B)
Definition	df-rdg 5344	Define a recursive definition generator on On (the class of ordinal numbers) with characteristic function F and initial value A. This combines functions F in tfr1 5336 and G in tz7.44-1 5340 into one definition. This rather amazing operation allows us to define, with compact direct definitions, functions that are usually defined in textbooks only with indirect self-referencing recursive definitions. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. On the other hand, the elimination of a direct definition is a matter of simple mechanical substitution. The price paid is the daunting complexity of our rec operation. But once we get past this hurdle, otherwise recursive definitions become relatively simple, as in for example oav 5401, from which we prove the recursive textbook definition as theorems oa0 5406, oasuc 5414, and oalim 5418 (with the help of theorems rdg0 5353, rdgsuc 5357, and rdglim2a 5362). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers om; see fr0g 5364 and frsuc 5365. Our rec operation apparently does not appear in published literature, although closely related is Definition 25.2 of [Quine] p. 177, which he uses to "turn...a recursion into a genuine or direct definition" (p. 174). Note that the if operations (see df-if 3213) select cases based on whether the domain of g is zero, a successor, or a limit ordinal. An important use of this definition is in the recursive sequence generator df-seq1 8210 on the natural numbers (as a subset of the complex numbers), allowing us to define, with direct definitions, recursive infinite sequences such as the factorial function df-fac 8685 and integer powers df-exp 8312. Note: We introduce rec with the philosophical goal of being able to eliminate all definitions with direct mechanical substitution and to verify easily the soundness of definitions. Metamath itself has no built-in technical limitation that prevents multiple-part recursive definitions in the traditional textbook style.
|- rec(F, A) = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | z = if(g = (/), A, if(Lim dom g, U.ran g, (F` (g` U.dom g))))}` (f |` y)))}
Theorem	dfrdg2 5345	Alternate definition of a recursive definition generator. (This was the original definition, but it was later replaced with the slightly shorter df-rdg 5344.)
|- rec(F, A) = U.{f | E.x e. On (f Fn x /\ A.y e. x (f` y) = ({<.g, z>. | ((g = (/) /\ z = A) \/ (-. (g = (/) \/ Lim dom g) /\ z = (F` (g` U.dom g))) \/ (Lim dom g /\ z = U.ran g))}` (f |` y)))}
Theorem	rdgeq1 5346	Equality theorem for the recursive definition generator.
|- (F = G -> rec(F, A) = rec(G, A))
Theorem	rdgeq2 5347	Equality theorem for the recursive definition generator.
|- (A = B -> rec(F, A) = rec(F, B))
Theorem	hbrdg 5348	Bound-variable hypothesis builder for the recursive definition generator.
|- (y e. F -> A.x y e. F)   &   |- (y e. A -> A.x y e. A)   =>   |- (y e. rec(F, A) -> A.x y e. rec(F, A))
Theorem	rdglem1 5349	Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use.
|- {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} = {g | E.z e. On (g Fn z /\ A.w e. z (g` w) = (G` (g |` w)))}
Theorem	rdglem2 5350	Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use.
|- {<.x, y>. | ((x = (/) /\ y = A) \/ (-. (x = (/) \/ Lim dom x) /\ y = (H` (x` U.dom x))) \/ (Lim dom x /\ y = U.ran x))} = {<.z, y>. | ((z = (/) /\ y = A) \/ (-. (z = (/) \/ Lim dom z) /\ y = (H` (z` U.dom z))) \/ (Lim dom z /\ y = U.ran z))}
Theorem	rdgfnon 5351	The recursive definition generator is a function on ordinal numbers.
|- rec(F, A) Fn On
Theorem	rdgval 5352	Value of the recursive definition generator.
|- (g e. On -> (rec(F, A)` g) = ({<.w, z>. | ((w = (/) /\ z = A) \/ (-. (w = (/) \/ Lim dom w) /\ z = (F` (w` U.dom w))) \/ (Lim dom w /\ z = U.ran w))}` (rec(F, A) |` g)))
Theorem	rdg0 5353	The initial value of the recursive definition generator.
|- A e. _V   =>   |- (rec(F, A)` (/)) = A
Theorem	rdgsuci 5354	The value of the recursive definition generator at a successor.
|- B e. On   =>   |- (rec(F, A)` suc B) = (F` (rec(F, A)` B))
Theorem	rdglimi 5355	The value of the recursive definition generator at a limit ordinal.
|- B e. On   =>   |- (Lim B -> (rec(F, A)` B) = U.(rec(F, A)"B))
Theorem	rdg0g 5356	The initial value of the recursive definition generator.
|- (A e. C -> (rec(F, A)` (/)) = A)
Theorem	rdgsuc 5357	The value of the recursive definition generator at a successor.
|- (B e. On -> (rec(F, A)` suc B) = (F` (rec(F, A)` B)))
Theorem	rdgsucopab 5358	The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered pair abstraction).
|- (z e. A -> A.x z e. A)   &   |- (z e. B -> A.x z e. B)   &   |- (z e. D -> A.x z e. D)   &   |- F = rec({<.x, y>. | y = C}, A)   &   |- (x = (F` B) -> C = D)   =>   |- ((B e. On /\ D e. R) -> (F` suc B) = D)
Theorem	rdgsucopabn 5359	The value of the recursive definition generator at a successor (special case where the characteristic function is an ordered-pair class abstraction and where the mapping class D is a proper class). This is a technical lemma that can be used together with rdgsucopab 5358 to help eliminate redundant sethood antecedents.
|- (z e. A -> A.x z e. A)   &   |- (z e. B -> A.x z e. B)   &   |- (z e. D -> A.x z e. D)   &   |- F = rec({<.x, y>. | y = C}, A)   &   |- (x = (F` B) -> C = D)   =>   |- (-. D e. _V -> (F` suc B) = (/))
Theorem	rdglim 5360	The value of the recursive definition generator at a limit ordinal.
|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.(rec(F, A)"B))
Theorem	rdglim2 5361	The value of the recursive definition generator at a limit ordinal, in terms of the union of all smaller values.
|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U.{y | E.x e. B y = (rec(F, A)` x)})
Theorem	rdglim2a 5362	The value of the recursive definition generator at a limit ordinal, in terms of indexed union of all smaller values.
|- ((B e. C /\ Lim B) -> (rec(F, A)` B) = U_x e. B (rec(F, A)` x))
Finite recursion
Theorem	frfnom 5363	The function generated by finite recursive definition generation is a function on omega.
|- (rec(F, A) |` om) Fn om
Theorem	fr0g 5364	The initial value resulting from finite recursive definition generation.
|- (A e. B -> ((rec(F, A) |` om)` (/)) = A)
Theorem	frsuc 5365	The successor value resulting from finite recursive definition generation.
|- (B e. om -> ((rec(F, A) |` om)` suc B) = (F` ((rec(F, A) |` om)` B)))
Theorem	frsucopab 5366	The successor value resulting from finite recursive definition generation (special case where the generation function is an ordered pair abstraction).
|- (z e. A -> A.x z e. A)   &   |- (z e. B -> A.x z e. B)   &   |- (z e. D -> A.x z e. D)   &   |- F = (rec({<.x, y>. | y = C}, A) |` om)   &   |- (x = (F` B) -> C = D)   =>   |- ((B e. om /\ D e. R) -> (F` suc B) = D)
Theorem	frsucmpt 5367	The successor value resulting from finite recursive definition generation (special case where the generation function is expressed in maps-to notation).
|- (z e. A -> A.x z e. A)   &   |- (z e. B -> A.x z e. B)   &   |- (z e. D -> A.x z e. D)   &   |- F = (rec((x e. _V |-> C), A) |` om)   &   |- (x = (F` B) -> C = D)   =>   |- ((B e. om /\ D e. R) -> (F` suc B) = D)
Theorem	tz7.48lem 5368	A way of showing an ordinal function is one-to-one.
|- F Fn On   =>   |- ((A C_ On /\ A.x e. A A.y e. x -. (F` x) = (F` y)) -> Fun `'(F |` A))
Theorem	tz7.48-2 5369	Proposition 7.48(2) of [TakeutiZaring] p. 51. (The proof was shortened by Mario Carneiro, 10-Jan-2013.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 5-May-2013.)
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)
Theorem	tz7.48-2OLD 5370	Obsolete proof of tz7.48-2 5369.
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)
Theorem	tz7.48-1 5371	Proposition 7.48(1) of [TakeutiZaring] p. 51.
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> ran F C_ A)
Theorem	tz7.48-3 5372	Proposition 7.48(3) of [TakeutiZaring] p. 51.
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> -. A e. _V)
Theorem	tz7.49OLD 5373	Proposition 7.49 of [TakeutiZaring] p. 51.
|- F Fn On   &   |- A e. _V   =>   |- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (A.y e. x (A \ (F"y)) =/= (/) /\ (F"x) = A /\ Fun `'(F |` x)))
Theorem	tz7.49cOLD 5374	Corollary of Proposition 7.49 of [TakeutiZaring] p. 51.
|- F Fn On   &   |- A e. _V   =>   |- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (F |` x):x-1-1-onto->A)
Theorem	tz7.49 5375	Proposition 7.49 of [TakeutiZaring] p. 51. (Revised by Mario Carneiro, 10-Jan-2013.)
|- F Fn On   &   |- (ph <-> A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))))   =>   |- ((A e. B /\ ph) -> E.x e. On (A.y e. x (A \ (F"y)) =/= (/) /\ (F"x) = A /\ Fun `'(F |` x)))
Theorem	tz7.49c 5376	Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Revised by Mario Carneiro, 19-Jan-2013.)
|- F Fn On   =>   |- ((A e. B /\ A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x)))) -> E.x e. On (F |` x):x-1-1-onto->A)
Abian's "most fundamental" fixed point theorem
Theorem	abianfplem 5377	Lemma for abianfp 5378. We prove by transfinite induction that if F has a fixed point x, then its iterates also equal x. This lemma is used for the "trivial" direction of the main theorem. [Auxiliary lemma - not displayed.]
Theorem	abianfp 5378	"A most fundamental fixed point theorem" of Alexander Abian (1923-1999), apparently proved in 1998. Let G` 0 = x, G` 1 = F` x, G` 2 = F` (F` x),... be the iterates of F. The theorem reads (using our variable names): "Let F be a mapping from a set A into itself. Then F has a fixed point if and only if: There exists an element x of A such that for every ordinal v, G` v is an element of A, and if G` v is not a fixed point of F then the G` u's are all distinct for every ordinal u e. v." See df-rdg 5344 for the rec operation. The proof's key idea is to assume that F does not have a fixed point, then use the Axiom of Replacement in the form of f1dmex 4789 to derive that the class of all ordinal numbers exists, contradicting onprc 4042. Our version of this theorem does not require the hypothesis that F be a mapping. Reference: http://us2.metamath.org:88/abian-themostfixed.html. For an application of this theorem, see http://groups.google.com/group/sci.stat.math/msg/1737ee1133c24aeb for its use in a proof of Tarski's fixed point theorem. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.)
|- A e. _V   &   |- G = rec({<.z, w>. | w = (F` z)}, x)   =>   |- (E.x e. A (F` x) = x <-> E.x e. A A.v e. On ((G` v) e. A /\ (-. (F` (G` v)) = (G` v) -> A.u e. v -. (G` v) = (G` u))))
Ordinal arithmetic
Syntax	c1o 5379	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 5380	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	coa 5381	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 5382	Extend the definition of a class to include the ordinal multiplication operation.
class .o
Syntax	coe 5383	Extend the definition of a class to include the ordinal exponentiation operation.
class ^o
Definition	df-1o 5384	Define the ordinal number 1.
|- 1o = suc (/)
Definition	df-2o 5385	Define the ordinal number 2.
|- 2o = suc 1o
Definition	df-oadd 5386	Define the ordinal addition operation.
|- +o = {<.<.x, y>., z>. | ((x e. On /\ y e. On) /\ z = (rec({<.w, v>. | v = suc w}, x)` y))}
Definition	df-omul 5387	Define the ordinal multiplication operation.
|- .o = {<.<.x, y>., z>. | ((x e. On /\ y e. On) /\ z = (rec({<.w, v>. | v = (w +o x)}, (/))` y))}
Definition	df-oexp 5388	Define the ordinal exponentiation operation.
|- ^o = {<.<.x, y>., z>. | ((x e. On /\ y e. On) /\ z = if(x = (/), (1o \ y), (rec({<.w, v>. | v = (w .o x)}, 1o)` y)))}
Theorem	1on 5389	Ordinal 1 is an ordinal number.
|- 1o e. On
Theorem	2on 5390	Ordinal 2 is an ordinal number. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- 2o e. On
Theorem	df1o2 5391	Expanded value of the ordinal number 1.
|- 1o = {(/)}
Theorem	df2o2 5392	Expanded value of the ordinal number 2.
|- 2o = {(/), {(/)}}
Theorem	1n0 5393	Ordinal one is not equal to ordinal zero.
|- 1o =/= (/)
Theorem	xp01disj 5394	Cross products with the singletons of ordinals 0 and 1 are disjoint.
|- ((A X. {(/)}) i^i (C X. {1o})) = (/)
Theorem	ordgt0ge1 5395	Two ways to express that an ordinal class is positive.
|- (Ord A -> ((/) e. A <-> 1o C_ A))
Theorem	ordge1n0 5396	An ordinal greater than or equal to 1 is nonzero.
|- (Ord A -> (1o C_ A <-> A =/= (/)))
Theorem	el1o 5397	Membership in ordinal one.
|- (A e. 1o <-> A = (/))
Theorem	0lt1o 5398	Ordinal zero is less than ordinal one.
|- (/) e. 1o
Theorem	fnoa 5399	Functionality and domain of ordinal addition.
|- +o Fn (On X. On)
Theorem	fnom 5400	Functionality and domain of ordinal multiplication.
|- .o Fn (On X. On)