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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdisjne 3301 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  C  e.  A  /\  D  e.  B ) 
 ->  C  =/=  D )
 
Theoremdisjel 3302 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  C  e.  A )  ->  -.  C  e.  B )
 
Theoremdisj2 3303 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
 |-  ( ( A  i^i  B )  =  (/)  <->  A  C_  ( _V  \  B ) )
 
Theoremdisj4im 3304 A consequence of two classes being disjoint. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 2-Aug-2018.)
 |-  ( ( A  i^i  B )  =  (/)  ->  -.  ( A  \  B )  C.  A )
 
Theoremssdisj 3305 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
 |-  ( ( A  C_  B  /\  ( B  i^i  C )  =  (/) )  ->  ( A  i^i  C )  =  (/) )
 
Theoremdisjpss 3306 A class is a proper subset of its union with a disjoint nonempty class. (Contributed by NM, 15-Sep-2004.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  B  =/=  (/) )  ->  A  C.  ( A  u.  B ) )
 
Theoremundisj1 3307 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
 |-  ( ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  C )  =  (/) )  <->  ( ( A  u.  B )  i^i 
 C )  =  (/) )
 
Theoremundisj2 3308 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
 |-  ( ( ( A  i^i  B )  =  (/)  /\  ( A  i^i  C )  =  (/) )  <->  ( A  i^i  ( B  u.  C ) )  =  (/) )
 
Theoremssindif0im 3309 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
 |-  ( A  C_  B  ->  ( A  i^i  ( _V  \  B ) )  =  (/) )
 
Theoreminelcm 3310 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
 |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( B  i^i  C )  =/=  (/) )
 
Theoremminel 3311 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
 |-  ( ( A  e.  B  /\  ( C  i^i  B )  =  (/) )  ->  -.  A  e.  C )
 
Theoremundif4 3312 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  C )  =  (/)  ->  ( A  u.  ( B  \  C ) )  =  ( ( A  u.  B )  \  C ) )
 
Theoremdisjssun 3313 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  B )  =  (/)  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C ) )
 
Theoremssdif0im 3314 Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
 |-  ( A  C_  B  ->  ( A  \  B )  =  (/) )
 
Theoremvdif0im 3315 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
 |-  ( A  =  _V  ->  ( _V  \  A )  =  (/) )
 
Theoremdifrab0eqim 3316* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)
 |-  ( V  =  { x  e.  V  |  ph
 }  ->  ( V  \  { x  e.  V  |  ph } )  =  (/) )
 
Theoremssnelpss 3317 A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
 |-  ( A  C_  B  ->  ( ( C  e.  B  /\  -.  C  e.  A )  ->  A  C.  B ) )
 
Theoremssnelpssd 3318 Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3317. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  C_  B )   &    |-  ( ph  ->  C  e.  B )   &    |-  ( ph  ->  -.  C  e.  A )   =>    |-  ( ph  ->  A  C.  B )
 
Theoreminssdif0im 3319 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
 |-  ( ( A  i^i  B )  C_  C  ->  ( A  i^i  ( B 
 \  C ) )  =  (/) )
 
Theoremdifid 3320 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
 |-  ( A  \  A )  =  (/)
 
TheoremdifidALT 3321 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3320. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A  \  A )  =  (/)
 
Theoremdif0 3322 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
 |-  ( A  \  (/) )  =  A
 
Theorem0dif 3323 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
 |-  ( (/)  \  A )  =  (/)
 
Theoremdisjdif 3324 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
 |-  ( A  i^i  ( B  \  A ) )  =  (/)
 
Theoremdifin0 3325 The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ( A  i^i  B )  \  B )  =  (/)
 
Theoremundif1ss 3326 Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( ( A  \  B )  u.  B )  C_  ( A  u.  B )
 
Theoremundif2ss 3327 Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( A  u.  ( B  \  A ) ) 
 C_  ( A  u.  B )
 
Theoremundifabs 3328 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
 |-  ( A  u.  ( A  \  B ) )  =  A
 
Theoreminundifss 3329 The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( ( A  i^i  B )  u.  ( A 
 \  B ) ) 
 C_  A
 
Theoremdifun2 3330 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
 |-  ( ( A  u.  B )  \  B )  =  ( A  \  B )
 
Theoremundifss 3331 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( A  C_  B  <->  ( A  u.  ( B 
 \  A ) ) 
 C_  B )
 
Theoremssdifin0 3332 A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
 |-  ( A  C_  ( B  \  C )  ->  ( A  i^i  C )  =  (/) )
 
Theoremssdifeq0 3333 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
 |-  ( A  C_  ( B  \  A )  <->  A  =  (/) )
 
Theoremssundifim 3334 A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( A  C_  ( B  u.  C )  ->  ( A  \  B ) 
 C_  C )
 
Theoremdifdifdirss 3335 Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( ( A  \  B )  \  C ) 
 C_  ( ( A 
 \  C )  \  ( B  \  C ) )
 
Theoremuneqdifeqim 3336 Two ways that  A and  B can "partition"  C (when  A and  B don't overlap and  A is a part of  C). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
 |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  ( ( A  u.  B )  =  C  ->  ( C  \  A )  =  B )
 )
 
Theoremr19.2m 3337* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1545). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
 
Theoremr19.3rm 3338* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
 |- 
 F/ x ph   =>    |-  ( E. y  y  e.  A  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.28m 3339* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |- 
 F/ x ph   =>    |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.3rmv 3340* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
 |-  ( E. y  y  e.  A  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.9rmv 3341* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |-  ( E. y  y  e.  A  ->  ( ph 
 <-> 
 E. x  e.  A  ph ) )
 
Theoremr19.28mv 3342* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.45mv 3343* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 |-  ( E. x  x  e.  A  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  (
 ph  \/  E. x  e.  A  ps ) ) )
 
Theoremr19.27m 3344* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |- 
 F/ x ps   =>    |-  ( E. x  x  e.  A  ->  (
 A. x  e.  A  ( ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  ps ) ) )
 
Theoremr19.27mv 3345* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  ps ) ) )
 
Theoremrzal 3346* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  (/)  ->  A. x  e.  A  ph )
 
Theoremrexn0 3347* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3348). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
 
Theoremrexm 3348* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
 |-  ( E. x  e.  A  ph  ->  E. x  x  e.  A )
 
Theoremralidm 3349* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
 |-  ( A. x  e.  A  A. x  e.  A  ph  <->  A. x  e.  A  ph )
 
Theoremral0 3350 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
 |- 
 A. x  e.  (/)  ph
 
Theoremrgenm 3351* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
 |-  ( ( E. x  x  e.  A  /\  x  e.  A )  -> 
 ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremralf0 3352* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
 |- 
 -.  ph   =>    |-  ( A. x  e.  A  ph  <->  A  =  (/) )
 
Theoremralm 3353 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
 |-  ( ( E. x  x  e.  A  ->  A. x  e.  A  ph ) 
 <-> 
 A. x  e.  A  ph )
 
Theoremraaanlem 3354* Special case of raaan 3355 where  A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) ) )
 
Theoremraaan 3355* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x  e.  A  A. y  e.  A  (
 ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  A. y  e.  A  ps ) )
 
Theoremraaanv 3356* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
 |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) )
 
Theoremsbss 3357* Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
 |-  ( [ y  /  x ] x  C_  A  <->  y 
 C_  A )
 
Theoremsbcssg 3358 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
 
2.1.15  Conditional operator
 
Syntaxcif 3359 Extend class notation to include the conditional operator. See df-if 3360 for a description. (In older databases this was denoted "ded".)
 class  if ( ph ,  A ,  B )
 
Definitiondf-if 3360* Define the conditional operator. Read  if ( ph ,  A ,  B ) as "if  ph then  A else  B." See iftrue 3364 and iffalse 3367 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 the absence of excluded middle, this will tend to be useful where  ph is decidable (in the sense of df-dc 754). (Contributed by NM, 15-May-1999.)

 |- 
 if ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
 
Theoremdfif6 3361* An alternate definition of the conditional operator df-if 3360 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 if ( ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )
 
Theoremifeq1 3362 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C )
 )
 
Theoremifeq2 3363 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B )
 )
 
Theoremiftrue 3364 Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
 
Theoremiftruei 3365 Inference associated with iftrue 3364. (Contributed by BJ, 7-Oct-2018.)
 |-  ph   =>    |- 
 if ( ph ,  A ,  B )  =  A
 
Theoremiftrued 3366 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ch )   =>    |-  ( ph  ->  if ( ch ,  A ,  B )  =  A )
 
Theoremiffalse 3367 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
 |-  ( -.  ph  ->  if ( ph ,  A ,  B )  =  B )
 
Theoremiffalsei 3368 Inference associated with iffalse 3367. (Contributed by BJ, 7-Oct-2018.)
 |- 
 -.  ph   =>    |- 
 if ( ph ,  A ,  B )  =  B
 
Theoremiffalsed 3369 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  if ( ch ,  A ,  B )  =  B )
 
Theoremifnefalse 3370 When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3367 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  =/=  B  ->  if ( A  =  B ,  C ,  D )  =  D )
 
Theoremdfif3 3371* Alternate definition of the conditional operator df-if 3360. Note that  ph is independent of  x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )
 
Theoremifeq12 3372 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D ) )
 
Theoremifeq1d 3373 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2d 3374 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifeq12d 3375 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D ) )
 
Theoremifbi 3376 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
 |-  ( ( ph  <->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
 
Theoremifbid 3377 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B ) )
 
Theoremifbieq1d 3378 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  B ,  C ) )
 
Theoremifbieq2i 3379 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 3380 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 3381 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
 |-  ( ph  <->  ps )   &    |-  A  =  C   &    |-  B  =  D   =>    |- 
 if ( ph ,  A ,  B )  =  if ( ps ,  C ,  D )
 
Theoremifbieq12d 3382 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 ) )
 
Theoremnfifd 3383 Deduction version of nfif 3384. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x if ( ps ,  A ,  B ) )
 
Theoremnfif 3384 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x if ( ph ,  A ,  B )
 
Theoremifbothdc 3385 A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by Jim Kingdon, 8-Aug-2021.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   =>    |-  ( ( ps 
 /\  ch  /\ DECID  ph )  ->  th )
 
Theoremifcldcd 3386 Membership (closure) of a conditional operator, deduction form. (Contributed by Jim Kingdon, 8-Aug-2021.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  C )   &    |-  ( ph  -> DECID  ps )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
 
2.1.16  Power classes
 
Syntaxcpw 3387 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
 class  ~P A
 
Theorempwjust 3388* Soundness justification theorem for df-pw 3389. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  C_  A }  =  {
 y  |  y  C_  A }
 
Definitiondf-pw 3389* 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. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if  A is { 3 , 5 , 7 }, then 
~P A is { (/) , { 3 } , { 5 } , { 7 } , { 3 , 5 } , { 3 , 7 } , { 5 , 7 } , { 3 , 5 , 7 } }. We will later introduce the Axiom of Power Sets. Still later we will prove that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 5-Aug-1993.)
 |- 
 ~P A  =  { x  |  x  C_  A }
 
Theorempweq 3390 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ~P A  =  ~P B )
 
Theorempweqi 3391 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
 |-  A  =  B   =>    |-  ~P A  =  ~P B
 
Theorempweqd 3392 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ~P A  =  ~P B )
 
Theoremelpw 3393 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  ~P B  <->  A  C_  B )
 
Theoremselpw 3394* Setvar variable membership in a power class (common case). See elpw 3393. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( x  e.  ~P A 
 <->  x  C_  A )
 
Theoremelpwg 3395 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
 |-  ( A  e.  V  ->  ( A  e.  ~P B 
 <->  A  C_  B )
 )
 
Theoremelpwi 3396 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  ~P B  ->  A  C_  B )
 
Theoremelpwid 3397 An element of a power class is a subclass. Deduction form of elpwi 3396. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ~P B )   =>    |-  ( ph  ->  A 
 C_  B )
 
Theoremelelpwi 3398 If  A belongs to a part of  C then  A belongs to  C. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  e.  B  /\  B  e.  ~P C )  ->  A  e.  C )
 
Theoremnfpw 3399 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x ~P A
 
Theorempwidg 3400 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  ~P A )
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