mmtheorems33 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3201-3300 - Page 33 of 195

Type	Label	Description
Statement
Theorem	r19.36zv 3201	Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty.
|- (A =/= (/) -> (E.x e. A (ph -> ps) <-> (A.x e. A ph -> ps)))
Theorem	rzal 3202	Vacuous quantification is always true. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
|- (A = (/) -> A.x e. A ph)
Theorem	rexn0 3203	Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- (E.x e. A ph -> A =/= (/))
Theorem	ralidm 3204	Idempotent law for restricted quantifier.
|- (A.x e. A A.x e. A ph <-> A.x e. A ph)
Theorem	ral0 3205	Vacuous universal quantification is always true.
|- A.x e. (/) ph
Theorem	rgenz 3206	Generalization rule that eliminates a non-zero class requirement.
|- ((A =/= (/) /\ x e. A) -> ph)   =>   |- A.x e. A ph
Theorem	ralf0 3207	The quantification of a falsehood is vacuous when true.
|- -. ph   =>   |- (A.x e. A ph <-> A = (/))
Theorem	raaan 3208	Rearrange restricted quantifiers.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   =>   |- (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps))
Theorem	raaanv 3209	Rearrange restricted quantifiers.
|- (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps))
Theorem	sbsslem 3210	Lemma for sbss 3211. (The proof was shortened by Andrew Salmon, 29-Jun-2011.) [Auxiliary lemma - not displayed.]
Theorem	sbss 3211	Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 26-Jun-2011.)
|- ([y / x]x C_ A <-> y C_ A)
"Weak deduction theorem" for set theory
Syntax	cif 3212	Extend class notation to include the conditional operator. See df-if 3213 for a description. (In older databases this was denoted "ded".)
class if(ph, A, B)
Definition	df-if 3213	Define the conditional operator. Read if(ph, A, B) as "if ph then A else B." See iftrue 3217 and iffalse 3218 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise." (In older versions of this database, this operator was denoted "ded" and called the "deduction class.") An important use for us is in conjunction with the weak deduction theorem, which converts a hypothesis into an antecedent. In that role, A is a class variable in the hypothesis and B is a class (usually a constant) that makes the hypothesis true when it is substituted for A. See dedth 3240 for the main part of the weak deduction theorem, elimhyp 3247 to eliminate a hypothesis, and keephyp 3253 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem.
|- if(ph, A, B) = {x | ((x e. A /\ ph) \/ (x e. B /\ -. ph))}
Theorem	dfif2 3214	An alternate definition of the conditional operator df-if 3213 with one fewer connectives (but probably less intuitive to understand).
|- if(ph, A, B) = {x | ((x e. B -> ph) -> (x e. A /\ ph))}
Theorem	ifeq1 3215	Equality theorem for conditional operator. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
|- (A = B -> if(ph, A, C) = if(ph, B, C))
Theorem	ifeq2 3216	Equality theorem for conditional operator. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
|- (A = B -> if(ph, C, A) = if(ph, C, B))
Theorem	iftrue 3217	Value of the conditional operator when its first argument is true. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
|- (ph -> if(ph, A, B) = A)
Theorem	iffalse 3218	Value of the conditional operator when its first argument is false.
|- (-. ph -> if(ph, A, B) = B)
Theorem	ifeq12 3219	Equality theorem for conditional operators.
|- ((A = B /\ C = D) -> if(ph, A, C) = if(ph, B, D))
Theorem	ifeq1d 3220	Equality deduction for conditional operator.
|- (ph -> A = B)   =>   |- (ph -> if(ps, A, C) = if(ps, B, C))
Theorem	ifeq2d 3221	Equality deduction for conditional operator.
|- (ph -> A = B)   =>   |- (ph -> if(ps, C, A) = if(ps, C, B))
Theorem	ifbi 3222	Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
|- ((ph <-> ps) -> if(ph, A, B) = if(ps, A, B))
Theorem	ifbid 3223	Equivalence deduction for conditional operators.
|- (ph -> (ps <-> ch))   =>   |- (ph -> if(ps, A, B) = if(ch, A, B))
Theorem	ifbieq2i 3224	Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
|- (ph <-> ps)   &   |- A = B   =>   |- if(ph, C, A) = if(ps, C, B)
Theorem	ifbieq2d 3225	Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
|- (ph -> (ps <-> ch))   &   |- (ph -> A = B)   =>   |- (ph -> if(ps, C, A) = if(ch, C, B))
Theorem	ifbieq12i 3226	Equivalence deduction for conditional operators.
|- (ph <-> ps)   &   |- A = C   &   |- B = D   =>   |- if(ph, A, B) = if(ps, C, D)
Theorem	ifbieq12d 3227	Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- (ph -> (ps <-> ch))   &   |- (ph -> A = C)   &   |- (ph -> B = D)   =>   |- (ph -> if(ps, A, B) = if(ch, C, D))
Theorem	hbif 3228	Bound-variable hypothesis builder for a conditional operator. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
|- (ph -> A.xph)   &   |- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   =>   |- (y e. if(ph, A, B) -> A.x y e. if(ph, A, B))
Theorem	hbifd 3229	Deduction version of hbif 3228.
|- (ph -> A.xph)   &   |- (ph -> (ps -> A.xps))   &   |- (ph -> (y e. A -> A.x y e. A))   &   |- (ph -> (y e. B -> A.x y e. B))   =>   |- (ph -> (y e. if(ps, A, B) -> A.x y e. if(ps, A, B)))
Theorem	csbifg 3230	Distribute proper substitution through the conditional operator.
|- (A e. V -> [_A / x]_if(ph, B, C) = if([A / x]ph, [_A / x]_B, [_A / x]_C))
Theorem	elimif 3231	Elimination of a conditional operator contained in a wff ps.
|- (if(ph, A, B) = A -> (ps <-> ch))   &   |- (if(ph, A, B) = B -> (ps <-> th))   =>   |- (ps <-> ((ph /\ ch) \/ (-. ph /\ th)))
Theorem	ifboth 3232	A wff th containing a conditional operator is true when both of its cases are true.
|- (A = if(ph, A, B) -> (ps <-> th))   &   |- (B = if(ph, A, B) -> (ch <-> th))   =>   |- ((ps /\ ch) -> th)
Theorem	ifid 3233	Identical true and false arguments in the conditional operator.
|- if(ph, A, A) = A
Theorem	eqif 3234	Expansion of an equality with a conditional operator.
|- (A = if(ph, B, C) <-> ((ph /\ A = B) \/ (-. ph /\ A = C)))
Theorem	elif 3235	Membership in a conditional operator.
|- (A e. if(ph, B, C) <-> ((ph /\ A e. B) \/ (-. ph /\ A e. C)))
Theorem	ifel 3236	Membership of a conditional operator.
|- (if(ph, A, B) e. C <-> ((ph /\ A e. C) \/ (-. ph /\ B e. C)))
Theorem	ifcl 3237	Membership (closure) of a conditional operator.
|- ((A e. C /\ B e. C) -> if(ph, A, B) e. C)
Theorem	ifor 3238	The possible values of a conditional operator. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
|- (if(ph, A, B) = A \/ if(ph, A, B) = B)
Theorem	ifswap 3239	Negating the first argument swaps the last two arguments of a conditional operator.
|- if(-. ph, A, B) = if(ph, B, A)
Theorem	dedth 3240	Weak deduction theorem that eliminates a hypothesis ph, making it become an antecedent. We assume that a proof exists for ph when the class variable A is replaced with a specific class B. The hypothesis ch should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3247. If the inference has other hypotheses with class variable A, these can be kept by assigning keephyp 3253 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpegif/mmdeduction.html.
|- (A = if(ph, A, B) -> (ps <-> ch))   &   |- ch   =>   |- (ph -> ps)
Theorem	dedth2h 3241	Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3244 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3240.
|- (A = if(ph, A, C) -> (ch <-> th))   &   |- (B = if(ps, B, D) -> (th <-> ta))   &   |- ta   =>   |- ((ph /\ ps) -> ch)
Theorem	dedth3h 3242	Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3241.
|- (A = if(ph, A, D) -> (th <-> ta))   &   |- (B = if(ps, B, R) -> (ta <-> et))   &   |- (C = if(ch, C, S) -> (et <-> ze))   &   |- ze   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	dedth4h 3243	Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3241.
|- (A = if(ph, A, R) -> (ta <-> et))   &   |- (B = if(ps, B, S) -> (et <-> ze))   &   |- (C = if(ch, C, F) -> (ze <-> si))   &   |- (D = if(th, D, G) -> (si <-> rh))   &   |- rh   =>   |- (((ph /\ ps) /\ (ch /\ th)) -> ta)
Theorem	dedth2v 3244	Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3241 is simpler to use. See also comments in dedth 3240. (The proof was shortened by Eric Schmidt, 28-Jul-2009.)
|- (A = if(ph, A, C) -> (ps <-> ch))   &   |- (B = if(ph, B, D) -> (ch <-> th))   &   |- th   =>   |- (ph -> ps)
Theorem	dedth3v 3245	Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3244. (The proof was shortened by Eric Schmidt, 28-Jul-2009.)
|- (A = if(ph, A, D) -> (ps <-> ch))   &   |- (B = if(ph, B, R) -> (ch <-> th))   &   |- (C = if(ph, C, S) -> (th <-> ta))   &   |- ta   =>   |- (ph -> ps)
Theorem	dedth4v 3246	Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3244. (The proof was shortened by Eric Schmidt, 28-Jul-2009.)
|- (A = if(ph, A, R) -> (ps <-> ch))   &   |- (B = if(ph, B, S) -> (ch <-> th))   &   |- (C = if(ph, C, T) -> (th <-> ta))   &   |- (D = if(ph, D, U) -> (ta <-> et))   &   |- et   =>   |- (ph -> ps)
Theorem	elimhyp 3247	Eliminate a hypothesis containing class variable A when it is known for a specific class B. For more information, see comments in dedth 3240.
|- (A = if(ph, A, B) -> (ph <-> ps))   &   |- (B = if(ph, A, B) -> (ch <-> ps))   &   |- ch   =>   |- ps
Theorem	elimhyp2v 3248	Eliminate a hypothesis containing 2 class variables.
|- (A = if(ph, A, C) -> (ph <-> ch))   &   |- (B = if(ph, B, D) -> (ch <-> th))   &   |- (C = if(ph, A, C) -> (ta <-> et))   &   |- (D = if(ph, B, D) -> (et <-> th))   &   |- ta   =>   |- th
Theorem	elimhyp3v 3249	Eliminate a hypothesis containing 3 class variables.
|- (A = if(ph, A, D) -> (ph <-> ch))   &   |- (B = if(ph, B, R) -> (ch <-> th))   &   |- (C = if(ph, C, S) -> (th <-> ta))   &   |- (D = if(ph, A, D) -> (et <-> ze))   &   |- (R = if(ph, B, R) -> (ze <-> si))   &   |- (S = if(ph, C, S) -> (si <-> ta))   &   |- et   =>   |- ta
Theorem	elimhyp4v 3250	Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3240).
|- (A = if(ph, A, D) -> (ph <-> ch))   &   |- (B = if(ph, B, R) -> (ch <-> th))   &   |- (C = if(ph, C, S) -> (th <-> ta))   &   |- (F = if(ph, F, G) -> (ta <-> ps))   &   |- (D = if(ph, A, D) -> (et <-> ze))   &   |- (R = if(ph, B, R) -> (ze <-> si))   &   |- (S = if(ph, C, S) -> (si <-> rh))   &   |- (G = if(ph, F, G) -> (rh <-> ps))   &   |- et   =>   |- ps
Theorem	elimel 3251	Eliminate a membership hypothesis for weak deduction theorem, when special case B e. C is provable.
|- B e. C   =>   |- if(A e. C, A, B) e. C
Theorem	elimdhyp 3252	Version of elimhyp 3247 where the hypothesis is deduced from the final antecedent. See ghomgrplem 14616 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
|- (ph -> ps)   &   |- (A = if(ph, A, B) -> (ps <-> ch))   &   |- (B = if(ph, A, B) -> (th <-> ch))   &   |- th   =>   |- ch
Theorem	keephyp 3253	Transform a hypothesis ps that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem.
|- (A = if(ph, A, B) -> (ps <-> th))   &   |- (B = if(ph, A, B) -> (ch <-> th))   &   |- ps   &   |- ch   =>   |- th
Theorem	keephyp2v 3254	Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3240).
|- (A = if(ph, A, C) -> (ps <-> ch))   &   |- (B = if(ph, B, D) -> (ch <-> th))   &   |- (C = if(ph, A, C) -> (ta <-> et))   &   |- (D = if(ph, B, D) -> (et <-> th))   &   |- ps   &   |- ta   =>   |- th
Theorem	keephyp3v 3255	Keep a hypothesis containing 3 class variables.
|- (A = if(ph, A, D) -> (rh <-> ch))   &   |- (B = if(ph, B, R) -> (ch <-> th))   &   |- (C = if(ph, C, S) -> (th <-> ta))   &   |- (D = if(ph, A, D) -> (et <-> ze))   &   |- (R = if(ph, B, R) -> (ze <-> si))   &   |- (S = if(ph, C, S) -> (si <-> ta))   &   |- rh   &   |- et   =>   |- ta
Theorem	keepel 3256	Keep a membership hypothesis for weak deduction theorem, when special case B e. C is provable.
|- A e. C   &   |- B e. C   =>   |- if(ph, A, B) e. C
Theorem	ifex 3257	Conditional operator existence.
|- A e. _V   &   |- B e. _V   =>   |- if(ph, A, B) e. _V
Theorem	ifexg 3258	Conditional operator existence.
|- ((A e. V /\ B e. W) -> if(ph, A, B) e. _V)
Power classes
Syntax	cpw 3259	Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
class ~PA
Theorem	pwjust 3260	Soundness justification theorem for df-pw 3261. (Contributed by Rodolfo Medina, 28-Apr-2010.) (The proof was shortened by Andrew Salmon, 29-Jun-2011.)
|- {x | x C_ A} = {y | y C_ A}
Definition	df-pw 3261	Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of _V.
|- ~PA = {x | x C_ A}
Theorem	pweq 3262	Equality theorem for the power class.
|- (A = B -> ~PA = ~PB)
Theorem	elpw 3263	Membership in a power class. Theorem 86 of [Suppes] p. 47.
|- A e. _V   =>   |- (A e. ~PB <-> A C_ B)
Theorem	elpwg 3264	Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 3662.
|- (A e. V -> (A e. ~PB <-> A C_ B))
Theorem	elpwi 3265	Subset relation implied by membership in a power class.
|- (A e. ~PB -> A C_ B)
Theorem	elelpwi 3266	If A belongs to a part of C then A belongs to C. (Contributed by FL, 3-Aug-2009.)
|- ((A e. B /\ B e. ~PC) -> A e. C)
Theorem	hbpw 3267	Bound-variable hypothesis builder for power class.
|- (y e. A -> A.x y e. A)   =>   |- (y e. ~PA -> A.x y e. ~PA)
Theorem	pwid 3268	A set is a member of its power class. Theorem 87 of [Suppes] p. 47.
|- A e. _V   =>   |- A e. ~PA
Theorem	pwss 3269	Subclass relationship for power class.
|- (~PA C_ B <-> A.x(x C_ A -> x e. B))
Unordered and ordered pairs
Syntax	csn 3270	Extend class notation to include singleton.
class {A}
Syntax	cpr 3271	Extend class notation to include unordered pair.
class {A, B}
Syntax	cop 3272	Extend class notation to include ordered pair.
class <.A, B>.
Theorem	snjust 3273	Soundness justification theorem for df-sn 3274. (Contributed by Rodolfo Medina, 28-Apr-2010.) (The proof was shortened by Andrew Salmon, 29-Jun-2011.)
|- {x | x = A} = {y | y = A}
Definition	df-sn 3274	Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of _V, although it is not very meaningful in this case. For an alternate definition see dfsn2 3282.
|- {A} = {x | x = A}
Definition	df-pr 3275	Define unordered pair of classes. Definition 7.1 of [Quine] p. 48. For a more traditional definition, but requiring a dummy variable, see dfpr2 3284.
|- {A, B} = ({A} u. {B})
Syntax	ctp 3276	Extend class notation to include unordered triplet.
class {A, B, C}
Definition	df-tp 3277	Define unordered triple of classes. Definition of [Enderton] p. 19.
|- {A, B, C} = ({A, B} u. {C})
Definition	df-op 3278	Kuratowski's ordered pair definition. Definition 9.1 of [Quine] p. 58. For proper classes it is not meaningful but is well-defined and we allow it for convenience (see opprc1 3391, opprc1b 3737, opprc2 3392, and opprc3 3738). For the justifying theorem (for sets) see opth 3727. There are other ways to define ordered pairs; the basic requirement is that two ordered pairs are equal iff their respective members are equal. In 1914 Norbert Wiener gave the first successful definition <.A, B>._2 = {{{A}, (/)}, {{B}}}, justified by opthwiener 3749, which was simplified by Kazimierz Kuratowski in 1921 to our present definition. An even simpler definition <.A, B>._3 = {A, {A, B}} is justified by opthreg 5985, but it requires the Axiom of Regularity for its justification and is not commonly used. A definition that also works for proper classes is <.A, B>._4 = ((A X. {(/)}) u. (B X. {{(/)}})), justified by opthprc 4211. If we restrict our sets to nonnegative integers, an ordered pair definition that involves only elementary arithmetic is provided by nn0opthi 8416. Finally, an ordered pair of real numbers can be represented by a complex number as shown by crui 8487.
|- <.A, B>. = {{A}, {A, B}}
Theorem	sneq 3279	Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15.
|- (A = B -> {A} = {B})
Theorem	sneqi 3280	Equality inference for singletons.
|- A = B   =>   |- {A} = {B}
Theorem	sneqd 3281	Equality deduction for singletons.
|- (ph -> A = B)   =>   |- (ph -> {A} = {B})
Theorem	dfsn2 3282	Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A} = {A, A}
Theorem	elsn 3283	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- (x e. {A} <-> x = A)
Theorem	dfpr2 3284	Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A, B} = {x | (x = A \/ x = B)}
Theorem	elprg 3285	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized.
|- (A e. V -> (A e. {B, C} <-> (A = B \/ A = C)))
Theorem	elpr 3286	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15.
|- A e. _V   =>   |- (A e. {B, C} <-> (A = B \/ A = C))
Theorem	elpr2 3287	A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15.
|- B e. _V   &   |- C e. _V   =>   |- (A e. {B, C} <-> (A = B \/ A = C))
Theorem	elpri 3288	If a class is an element of a pair, then it is one of the two paired elements. (Contributed by Scott Fenton, 1-Apr-2011.)
|- (A e. {B, C} -> (A = B \/ A = C))
Theorem	elsncg 3289	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (The proof was shortened by Andrew Salmon, 29-Jun-2011.)
|- (A e. V -> (A e. {B} <-> A = B))
Theorem	elsnc 3290	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- A e. _V   =>   |- (A e. {B} <-> A = B)
Theorem	elsni 3291	There is only one element in a singleton.
|- (A e. {B} -> A = B)
Theorem	snidg 3292	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.
|- (A e. V -> A e. {A})
Theorem	snidb 3293	A class is a set iff it is a member of its singleton.
|- (A e. _V <-> A e. {A})
Theorem	snid 3294	A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.
|- A e. _V   =>   |- A e. {A}
Theorem	elsnc2g 3295	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set.
|- (B e. V -> (A e. {B} <-> A = B))
Theorem	elsnc2 3296	There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set.
|- B e. _V   =>   |- (A e. {B} <-> A = B)
Theorem	rexsng 3297	Restricted existential quantification over a singleton.
|- (x = A -> (ph <-> ps))   =>   |- (A e. V -> (E.x e. {A}ph <-> ps))
Theorem	rexsn 3298	Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
|- A e. _V   &   |- (x = A -> (ph <-> ps))   =>   |- (E.x e. {A}ph <-> ps)
Theorem	eltp 3299	A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17.
|- A e. _V   =>   |- (A e. {B, C, D} <-> (A = B \/ A = C \/ A = D))
Theorem	dftp2 3300	Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16.
|- {A, B, C} = {x | (x = A \/ x = B \/ x = C)}