mmtheorems52 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 5101-5200 - Page 52 of 175

Type	Label	Description
Statement
Theorem	iota4an 5101	Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
|- (E!x(ph /\ ps) -> [(iotax(ph /\ ps)) / x]ph)
Theorem	iotabidv 5102	Formula-building deduction rule for iota.
|- (ph -> (ps <-> ch))   =>   |- (ph -> (iotaxps) = (iotaxch))
Theorem	iotacl 5103	Membership law for descriptions. (Contributed by Andrew Salmon, 1-Aug-2011.)
|- (E!xph -> (iotaxph) e. {x | ph})
Theorem	iota1 5104	Property of iota. Compare euuni 3807.
|- (E!xph -> (ph <-> (iotaxph) = x))
Theorem	reiotacl2 5105	Membership law for descriptions. Compare reucl2 3814.
|- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. {x e. A | ph})
Theorem	reiotacl 5106	Membership law for descriptions. Compare reucl 3213.
|- (E!x e. A ph -> (iotax(x e. A /\ ph)) e. A)
Theorem	reiota4 5107	Substitution law for descriptions. Compare reuuni4 3813.
|- (E!x e. A ph -> [(iotax(x e. A /\ ph)) / x]ph)
Theorem	reiota1 5108	Property of iota. Compare reuuni1 3808.
|- ((x e. A /\ E!x e. A ph) -> (ph <-> (iotax(x e. A /\ ph)) = x))
Theorem	reiota2f 5109	A condition that allows us to represent "the unique element in A such that ph " with a class expression B. The second hypothesis is a weakened bound variable condition that allows hbsbc1g 2461 to be used. Compare reuuni2f 3810.
|- (y e. B -> A.x y e. B)   &   |- (B e. A -> (ps -> A.xps))   &   |- (x = B -> (ph <-> ps))   =>   |- ((B e. A /\ E!x e. A ph) -> (ps <-> (iotax(x e. A /\ ph)) = B))
Theorem	reiota2 5110	A condition that allows us to represent "the unique element in A such that ph " with a class expression B. Compare reuuni2 3811.
|- (x = B -> (ph <-> ps))   =>   |- ((B e. A /\ E!x e. A ph) -> (ps <-> (iotax(x e. A /\ ph)) = B))
Theorem	reiotass2 5111	Restriction of a unique element to a smaller class. Compare reuuniss2 3817.
|- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (iotax(x e. A /\ ph)) = (iotax(x e. B /\ ps)))
Cantor's Theorem
Theorem	canth 5112	No set A is equinumerous to its power set (Cantor's theorem), i.e. no function can map A it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 5548. Note that A must be a set: this theorem does not hold when A is too large to be a set; see ncanth 5113 for a counterexample. (Use nex 1456 if you want the form -. E.ff:A-onto->~PA.)
|- A e. _V   =>   |- -. F:A-onto->~PA
Theorem	ncanth 5113	Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 3449). Specifically, the identity function maps the universe onto its power class. Compare canth 5112 that works for sets. See also the remark in ru 2451 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set.
|- _I :_V-onto->~P_V
Miscellaneous ordinal theorems (that depend on functions and relations)
Theorem	iunon 5114	The indexed union of a set of ordinal numbers B(x) is an ordinal number.
|- A e. _V   &   |- B e. _V   =>   |- (A.x e. A B e. On -> U_x e. A B e. On)
Theorem	iinon 5115	The nonempty indexed intersection of a class of ordinal numbers B(x) is an ordinal number.
|- B e. _V   =>   |- ((A.x e. A B e. On /\ A =/= (/)) -> |^|_x e. A B e. On)
Theorem	onfununi 5116	A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)
|- (Lim y -> (F` y) = U_x e. y (F` x))   &   |- ((x e. On /\ y e. On /\ x C_ y) -> (F` x) C_ (F` y))   =>   |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (F` U.S) = U_x e. S (F` x))
Theorem	onopruni 5117	A variant of onfununi 5116 for operations. (Contributed by Eric Schmidt, 26-May-2009.)
|- (Lim y -> (AFy) = U_x e. y (AFx))   &   |- ((x e. On /\ y e. On /\ x C_ y) -> (AFx) C_ (AFy))   =>   |- ((S e. T /\ S C_ On /\ S =/= (/)) -> (AFU.S) = U_x e. S (AFx))
Theorem	onopriun 5118	A variant of onopruni 5117 with indexed unions. (Contributed by Eric Schmidt, 26-May-2009.)
|- (Lim y -> (AFy) = U_x e. y (AFx))   &   |- ((x e. On /\ y e. On /\ x C_ y) -> (AFx) C_ (AFy))   =>   |- ((K e. T /\ A.z e. K L e. On /\ K =/= (/)) -> (AFU_z e. K L) = U_z e. K (AFL))
Transfinite recursion
Theorem	tfrlem1 5119	A technical lemma for transfinite recursion. Compare Lemma 1 of [TakeutiZaring] p. 47.
|- (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 5120	Lemma for transfinite recursion. This provides some messy details needed to link tfrlem1 5119 into the main proof.
Theorem	tfrlem3 5121	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.
Theorem	tfrlem4 5122	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.
Theorem	tfrlem5 5123	Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains.
Theorem	tfrlem6 5124	Lemma for transfinite recursion. The union of all acceptable functions is a relation.
Theorem	tfrlem7 5125	Lemma for transfinite recursion. The union of all acceptable functions is a function.
Theorem	tfrlem8 5126	Lemma for transfinite recursion. The domain of F is ordinal. (The proof was shortened by Alan Sare, 11-Mar-2008.)
Theorem	tfrlem9 5127	Lemma for transfinite recursion. Here we compute the value of F (the union of all acceptable functions).
Theorem	tfrlem10 5128	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 5131, 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.)
Theorem	tfrlem11 5129	Lemma for transfinite recursion. Compute the value of C.
Theorem	tfrlem12 5130	Lemma for transfinite recursion. Show C is an acceptable function.
Theorem	tfrlem13 5131	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.
Theorem	tfr1 5132	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 5133	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 5134	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 5135	G is a function. Lemma for tz7.44-1 5136, tz7.44-2 5137, and tz7.44-3 5138.
|- 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 5136	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 5137	The value of F at a successor ordinal. Part 2 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)))   &   |- B e. On   =>   |- (F` suc B) = (H` (F` B))
Theorem	tz7.44-3 5138	The value of F at a limit ordinal. Part 3 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)))   &   |- B e. On   =>   |- (Lim B -> (F` B) = U.(F"B))
Recursive definition generator
Syntax	crdg 5139	Extend class notation with the recursive definition generator.
class rec(A, B)
Definition	df-rdg 5140	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 5132 and G in tz7.44-1 5136 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 5195, from which we prove the recursive textbook definition as theorems oa0 5200, oasuc 5208, and oalim 5212 (with the help of theorems rdg0 5149, rdgsuc 5153, and rdglimi 5151). We can also restrict the rec operation to define otherwise recursive functions on the natural numbers om; see fr0g 5160 and frsuc 5161. 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 2983) 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 7721 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 8184 and integer powers df-exp 7812. 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 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 5141	Alternate definition of a recursive definition generator. (This was the original definition, but it was later replaced with the slightly shorter df-rdg 5140.)
|- 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 5142	Equality theorem for the recursive definition generator.
|- (F = G -> rec(F, A) = rec(G, A))
Theorem	rdgeq2 5143	Equality theorem for the recursive definition generator.
|- (A = B -> rec(F, A) = rec(F, B))
Theorem	hbrdg 5144	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 5145	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 5146	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 5147	The recursive definition generator is a function on ordinal numbers.
|- rec(F, A) Fn On
Theorem	rdgval 5148	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 5149	The initial value of the recursive definition generator.
|- A e. _V   =>   |- (rec(F, A)` (/)) = A
Theorem	rdgsuci 5150	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 5151	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 5152	The initial value of the recursive definition generator.
|- (A e. C -> (rec(F, A)` (/)) = A)
Theorem	rdgsuc 5153	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 5154	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 5155	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 5154 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 5156	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 5157	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 5158	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 5159	The function generated by finite recursive definition generation is a function on omega.
|- (rec(F, A) |` om) Fn om
Theorem	fr0g 5160	The initial value resulting from finite recursive definition generation.
|- (A e. B -> ((rec(F, A) |` om)` (/)) = A)
Theorem	frsuc 5161	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 5162	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 5163	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 5164	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	tz7.48-1 5165	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	tz7.48-2 5166	Proposition 7.48(2) of [TakeutiZaring] p. 51.
|- F Fn On   =>   |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)
Theorem	tz7.48-3 5167	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	tz7.49 5168	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	tz7.49c 5169	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)
Abian's "most fundamental" fixed point theorem
Theorem	abianfplem 5170	Lemma for abianfp 5171. 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.
Theorem	abianfp 5171	"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 5140 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 4664 to derive that the class of all ordinal numbers exists, contradicting onprc 3865. 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.
|- 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 5172	Extend the definition of a class to include the ordinal number 1.
class 1o
Syntax	c2o 5173	Extend the definition of a class to include the ordinal number 2.
class 2o
Syntax	coa 5174	Extend the definition of a class to include the ordinal addition operation.
class +o
Syntax	comu 5175	Extend the definition of a class to include the ordinal multiplication operation.
class .o
Syntax	coe 5176	Extend the definition of a class to include the ordinal exponentiation operation.
class ^o
Definition	df-1o 5177	Define the ordinal number 1.
|- 1o = suc (/)
Definition	df-2o 5178	Define the ordinal number 2.
|- 2o = suc 1o
Definition	df-oadd 5179	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 5180	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 5181	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 5182	Ordinal 1 is an ordinal number.
|- 1o e. On
Theorem	2on 5183	Ordinal 2 is an ordinal number. (The proof was shortened by Andrew Salmon, 12-Aug-2011.)
|- 2o e. On
Theorem	2onOLD 5184	Ordinal 2 is an ordinal number.
|- 2o e. On
Theorem	df1o2 5185	Expanded value of the ordinal number 1.
|- 1o = {(/)}
Theorem	df2o2 5186	Expanded value of the ordinal number 2.
|- 2o = {(/), {(/)}}
Theorem	1n0 5187	Ordinal one is not equal to ordinal zero.
|- 1o =/= (/)
Theorem	xp01disj 5188	Cross products with the singletons of ordinals 0 and 1 are disjoint.
|- ((A X. {(/)}) i^i (C X. {1o})) = (/)
Theorem	ordgt0ge1 5189	Two ways to express that an ordinal class is positive.
|- (Ord A -> ((/) e. A <-> 1o C_ A))
Theorem	ordge1n0 5190	An ordinal greater than or equal to 1 is nonzero.
|- (Ord A -> (1o C_ A <-> A =/= (/)))
Theorem	el1o 5191	Membership in ordinal one.
|- (A e. 1o <-> A = (/))
Theorem	0lt1o 5192	Ordinal zero is less than ordinal one.
|- (/) e. 1o
Theorem	fnoa 5193	Functionality and domain of ordinal addition.
|- +o Fn (On X. On)
Theorem	fnom 5194	Functionality and domain of ordinal multiplication.
|- .o Fn (On X. On)
Theorem	oav 5195	Value of ordinal addition.
|- ((A e. On /\ B e. On) -> (A +o B) = (rec({<.x, y>. | y = suc x}, A)` B))
Theorem	omv 5196	Value of ordinal multiplication.
|- ((A e. On /\ B e. On) -> (A .o B) = (rec({<.x, y>. | y = (x +o A)}, (/))` B))
Theorem	oe0lem 5197	A helper lemma for oe0 5206 and others.
|- ((ph /\ A = (/)) -> ps)   &   |- (((A e. On /\ ph) /\ (/) e. A) -> ps)   =>   |- ((A e. On /\ ph) -> ps)
Theorem	oev 5198	Value of ordinal exponentiation.
|- ((A e. On /\ B e. On) -> (A ^o B) = if(A = (/), (1o \ B), (rec({<.x, y>. | y = (x .o A)}, 1o)` B)))
Theorem	oevn0 5199	Value of ordinal exponentiation at a nonzero mantissa.
|- (((A e. On /\ B e. On) /\ (/) e. A) -> (A ^o B) = (rec({<.x, y>. | y = (x .o A)}, 1o)` B))
Theorem	oa0 5200	Addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57.
|- (A e. On -> (A +o (/)) = A)