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Statement List for Metamath Proof Explorer - 3201-3300 - Page 33 of 195
TypeLabelDescription
Statement
 
Theoremr19.36zv 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)))
 
Theoremrzal 3202 Vacuous quantification is always true. (The proof was shortened by Andrew Salmon, 26-Jun-2011.)
|- (A = (/) -> A.x e. A ph)
 
Theoremrexn0 3203 Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
|- (E.x e. A ph -> A =/= (/))
 
Theoremralidm 3204 Idempotent law for restricted quantifier.
|- (A.x e. A A.x e. A ph <-> A.x e. A ph)
 
Theoremral0 3205 Vacuous universal quantification is always true.
|- A.x e. (/) ph
 
Theoremrgenz 3206 Generalization rule that eliminates a non-zero class requirement.
|- ((A =/= (/) /\ x e. A) -> ph)   =>   |- A.x e. A ph
 
Theoremralf0 3207 The quantification of a falsehood is vacuous when true.
|- -. ph   =>   |- (A.x e. A ph <-> A = (/))
 
Theoremraaan 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))
 
Theoremraaanv 3209 Rearrange restricted quantifiers.
|- (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps))
 
Theoremsbsslem 3210 Lemma for sbss 3211. (The proof was shortened by Andrew Salmon, 29-Jun-2011.) [Auxiliary lemma - not displayed.]
 
Theoremsbss 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
 
Syntaxcif 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)
 
Definitiondf-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))}
 
Theoremdfif2 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))}
 
Theoremifeq1 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))
 
Theoremifeq2 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))
 
Theoremiftrue 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)
 
Theoremiffalse 3218 Value of the conditional operator when its first argument is false.
|- (-. ph -> if(ph, A, B) = B)
 
Theoremifeq12 3219 Equality theorem for conditional operators.
|- ((A = B /\ C = D) -> if(ph, A, C) = if(ph, B, D))
 
Theoremifeq1d 3220 Equality deduction for conditional operator.
|- (ph -> A = B)   =>   |- (ph -> if(ps, A, C) = if(ps, B, C))
 
Theoremifeq2d 3221 Equality deduction for conditional operator.
|- (ph -> A = B)   =>   |- (ph -> if(ps, C, A) = if(ps, C, B))
 
Theoremifbi 3222 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
|- ((ph <-> ps) -> if(ph, A, B) = if(ps, A, B))
 
Theoremifbid 3223 Equivalence deduction for conditional operators.
|- (ph -> (ps <-> ch))   =>   |- (ph -> if(ps, A, B) = if(ch, A, B))
 
Theoremifbieq2i 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)
 
Theoremifbieq2d 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))
 
Theoremifbieq12i 3226 Equivalence deduction for conditional operators.
|- (ph <-> ps)   &   |- A = C   &   |- B = D   =>   |- if(ph, A, B) = if(ps, C, D)
 
Theoremifbieq12d 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))
 
Theoremhbif 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))
 
Theoremhbifd 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)))
 
Theoremcsbifg 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))
 
Theoremelimif 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)))
 
Theoremifboth 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)
 
Theoremifid 3233 Identical true and false arguments in the conditional operator.
|- if(ph, A, A) = A
 
Theoremeqif 3234 Expansion of an equality with a conditional operator.
|- (A = if(ph, B, C) <-> ((ph /\ A = B) \/ (-. ph /\ A = C)))
 
Theoremelif 3235 Membership in a conditional operator.
|- (A e. if(ph, B, C) <-> ((ph /\ A e. B) \/ (-. ph /\ A e. C)))
 
Theoremifel 3236 Membership of a conditional operator.
|- (if(ph, A, B) e. C <-> ((ph /\ A e. C) \/ (-. ph /\ B e. C)))
 
Theoremifcl 3237 Membership (closure) of a conditional operator.
|- ((A e. C /\ B e. C) -> if(ph, A, B) e. C)
 
Theoremifor 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)
 
Theoremifswap 3239 Negating the first argument swaps the last two arguments of a conditional operator.
|- if(-. ph, A, B) = if(ph, B, A)
 
Theoremdedth 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)
 
Theoremdedth2h 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)
 
Theoremdedth3h 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)
 
Theoremdedth4h 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)
 
Theoremdedth2v 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)
 
Theoremdedth3v 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)
 
Theoremdedth4v 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)
 
Theoremelimhyp 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
 
Theoremelimhyp2v 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
 
Theoremelimhyp3v 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
 
Theoremelimhyp4v 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
 
Theoremelimel 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
 
Theoremelimdhyp 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
 
Theoremkeephyp 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
 
Theoremkeephyp2v 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
 
Theoremkeephyp3v 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
 
Theoremkeepel 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
 
Theoremifex 3257 Conditional operator existence.
|- A e. _V   &   |- B e. _V   =>   |- if(ph, A, B) e. _V
 
Theoremifexg 3258 Conditional operator existence.
|- ((A e. V /\ B e. W) -> if(ph, A, B) e. _V)
 
Power classes
 
Syntaxcpw 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
 
Theorempwjust 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}
 
Definitiondf-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}
 
Theorempweq 3262 Equality theorem for the power class.
|- (A = B -> ~PA = ~PB)
 
Theoremelpw 3263 Membership in a power class. Theorem 86 of [Suppes] p. 47.
|- A e. _V   =>   |- (A e. ~PB <-> A C_ B)
 
Theoremelpwg 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))
 
Theoremelpwi 3265 Subset relation implied by membership in a power class.
|- (A e. ~PB -> A C_ B)
 
Theoremelelpwi 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)
 
Theoremhbpw 3267 Bound-variable hypothesis builder for power class.
|- (y e. A -> A.x y e. A)   =>   |- (y e. ~PA -> A.x y e. ~PA)
 
Theorempwid 3268 A set is a member of its power class. Theorem 87 of [Suppes] p. 47.
|- A e. _V   =>   |- A e. ~PA
 
Theorempwss 3269 Subclass relationship for power class.
|- (~PA C_ B <-> A.x(x C_ A -> x e. B))
 
Unordered and ordered pairs
 
Syntaxcsn 3270 Extend class notation to include singleton.
class {A}
 
Syntaxcpr 3271 Extend class notation to include unordered pair.
class {A, B}
 
Syntaxcop 3272 Extend class notation to include ordered pair.
class <.A, B>.
 
Theoremsnjust 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}
 
Definitiondf-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}
 
Definitiondf-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})
 
Syntaxctp 3276 Extend class notation to include unordered triplet.
class {A, B, C}
 
Definitiondf-tp 3277 Define unordered triple of classes. Definition of [Enderton] p. 19.
|- {A, B, C} = ({A, B} u. {C})
 
Definitiondf-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}}
 
Theoremsneq 3279 Equality theorem for singletons. Part of Exercise 4 of [TakeutiZaring] p. 15.
|- (A = B -> {A} = {B})
 
Theoremsneqi 3280 Equality inference for singletons.
|- A = B   =>   |- {A} = {B}
 
Theoremsneqd 3281 Equality deduction for singletons.
|- (ph -> A = B)   =>   |- (ph -> {A} = {B})
 
Theoremdfsn2 3282 Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A} = {A, A}
 
Theoremelsn 3283 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- (x e. {A} <-> x = A)
 
Theoremdfpr2 3284 Alternate definition of unordered pair. Definition 5.1 of [TakeutiZaring] p. 15.
|- {A, B} = {x | (x = A \/ x = B)}
 
Theoremelprg 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)))
 
Theoremelpr 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))
 
Theoremelpr2 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))
 
Theoremelpri 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))
 
Theoremelsncg 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))
 
Theoremelsnc 3290 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15.
|- A e. _V   =>   |- (A e. {B} <-> A = B)
 
Theoremelsni 3291 There is only one element in a singleton.
|- (A e. {B} -> A = B)
 
Theoremsnidg 3292 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.
|- (A e. V -> A e. {A})
 
Theoremsnidb 3293 A class is a set iff it is a member of its singleton.
|- (A e. _V <-> A e. {A})
 
Theoremsnid 3294 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49.
|- A e. _V   =>   |- A e. {A}
 
Theoremelsnc2g 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))
 
Theoremelsnc2 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)
 
Theoremrexsng 3297 Restricted existential quantification over a singleton.
|- (x = A -> (ph <-> ps))   =>   |- (A e. V -> (E.x e. {A}ph <-> ps))
 
Theoremrexsn 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)
 
Theoremeltp 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))
 
Theoremdftp2 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)}

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