HomeHome Metamath Proof Explorer
Theorem List (p. 36 of 314)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21444)
  Hilbert Space Explorer  Hilbert Space Explorer
(21445-22967)
  Users' Mathboxes  Users' Mathboxes
(22968-31305)
 

Theorem List for Metamath Proof Explorer - 3501-3600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempssdifcom2 3501 Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
 |-  ( ( A  C_  C  /\  B  C_  C )  ->  ( B  C.  ( C  \  A )  <->  A  C.  ( C  \  B ) ) )
 
Theoremdifdifdir 3502 Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
 |-  ( ( A  \  B )  \  C )  =  ( ( A 
 \  C )  \  ( B  \  C ) )
 
Theoremuneqdifeq 3503 Two ways to say that  A and  B partition  C (when  A and  B don't overlap and  A is a part of  C). (Contributed by FL, 17-Nov-2008.)
 |-  ( ( A  C_  C  /\  ( A  i^i  B )  =  (/) )  ->  ( ( A  u.  B )  =  C  <->  ( C  \  A )  =  B ) )
 
Theoremr19.2z 3504* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1759). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
 |-  ( ( A  =/=  (/)  /\  A. x  e.  A  ph )  ->  E. x  e.  A  ph )
 
Theoremr19.2zb 3505* A response to the notion that the condition  A  =/=  (/) can be removed in r19.2z 3504. Interestingly enough,  ph does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
 |-  ( A  =/=  (/)  <->  ( A. x  e.  A  ph  ->  E. x  e.  A  ph ) )
 
Theoremr19.3rz 3506* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
 |- 
 F/ x ph   =>    |-  ( A  =/=  (/)  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.28z 3507* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
 |- 
 F/ x ph   =>    |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.3rzv 3508* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
 |-  ( A  =/=  (/)  ->  ( ph 
 <-> 
 A. x  e.  A  ph ) )
 
Theoremr19.9rzv 3509* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
 |-  ( A  =/=  (/)  ->  ( ph 
 <-> 
 E. x  e.  A  ph ) )
 
Theoremr19.28zv 3510* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)
 |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 ph  /\  A. x  e.  A  ps ) ) )
 
Theoremr19.37zv 3511* Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)
 |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  (
 ph  ->  E. x  e.  A  ps ) ) )
 
Theoremr19.45zv 3512* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
 |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  (
 ph  \/  E. x  e.  A  ps ) ) )
 
Theoremr19.27z 3513* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
 |- 
 F/ x ps   =>    |-  ( A  =/=  (/) 
 ->  ( A. x  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  ps )
 ) )
 
Theoremr19.27zv 3514* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)
 |-  ( A  =/=  (/)  ->  ( A. x  e.  A  ( ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  ps ) ) )
 
Theoremr19.36zv 3515* Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)
 |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  ->  ps )  <->  (
 A. x  e.  A  ph 
 ->  ps ) ) )
 
Theoremrzal 3516* 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 3517* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
 
Theoremralidm 3518* 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 3519 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
 |- 
 A. x  e.  (/)  ph
 
Theoremrgenz 3520* Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)
 |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremralf0 3521* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
 |- 
 -.  ph   =>    |-  ( A. x  e.  A  ph  <->  A  =  (/) )
 
Theoremraaan 3522* 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 3523* 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 3524* 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 )
 
2.1.15  "Weak deduction theorem" for set theory

In a Hilbert system of logic (which consists of a set of axioms, modus ponens, and the generalization rule), converting a deduction to a proof using the Deduction Theorem (taught in introductory logic books) involves an exponential increase of the number of steps as hypotheses are successively eliminated. Here is a trick that is not as general as the Deduction Theorem but requires only a linear increase in the number of steps.

The general problem: We want to convert a deduction P |- Q into a proof of the theorem |- P -> Q i.e. we want to eliminate the hypothesis P. Normally this is done using the Deduction (meta)Theorem, which looks at the microscopic steps of the deduction and usually doubles or triples the number of these microscopic steps for each hypothesis that is eliminated. We will look at a special case of this problem, without appealing to the Deduction Theorem.

We assume ZF with class notation. A and B are arbitrary (possibly proper) classes. P, Q, R, S and T are wffs.

We define the conditional operator, if(P,A,B), as follows: if(P,A,B) =def= { x | (x \in A & P) v (x \in B & -. P) } (where x does not occur in A, B, or P).

Lemma 1. A = if(P,A,B) -> (P <-> R), B = if(P,A,B) -> (S <-> R), S |- R Proof: Logic and Axiom of Extensionality.

Lemma 2. A = if(P,A,B) -> (Q <-> T), T |- P -> Q Proof: Logic and Axiom of Extensionality.

Here's a simple example that illustrates how it works. Suppose we have a deduction Ord A |- Tr A which means, "Assume A is an ordinal class. Then A is a transitive class." Note that A is a class variable that may be substituted with any class expression, so this is really a deduction scheme.

We want to convert this to a proof of the theorem (scheme) |- Ord A -> Tr A.

The catch is that we must be able to prove "Ord A" for at least one object A (and this is what makes it weaker than the ordinary Deduction Theorem). However it is easy to prove |- Ord 0 (the empty set is ordinal). (For a typical textbook "theorem," i.e. deduction, there is usually at least one object satisfying each hypothesis, otherwise the theorem would not be very useful. We can always go back to the standard Deduction Theorem for those hypotheses where this is not the case.) Continuing with the example:

Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0)) (1) |- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0)) (2) From (1), (2) and |- Ord 0, Lemma 1 yields |- Ord if(Ord A, A, 0) (3) From (3) and substituting if(Ord A, A, 0) for A in the original deduction, |- Tr if(Ord A, A, 0) (4) Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0)) (5) From (4) and (5), Lemma 2 yields |- Ord A -> Tr A (Q.E.D.)

 
Syntaxcif 3525 Extend class notation to include the conditional operator. See df-if 3526 for a description. (In older databases this was denoted "ded".)
 class  if ( ph ,  A ,  B )
 
Definitiondf-if 3526* Define the conditional operator. Read  if ( ph ,  A ,  B ) as "if  ph then  A else  B." See iftrue 3531 and iffalse 3532 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 3566 for the main part of the weak deduction theorem, elimhyp 3573 to eliminate a hypothesis, and keephyp 3579 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem. (Contributed by NM, 15-May-1999.)

 |- 
 if ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
 
Theoremdfif2 3527* An alternate definition of the conditional operator df-if 3526 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
 |- 
 if ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  B  -> 
 ph )  ->  ( x  e.  A  /\  ph ) ) }
 
Theoremdfif6 3528* An alternate definition of the conditional operator df-if 3526 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 3529 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 3530 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 3531 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 )
 
Theoremiffalse 3532 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
 |-  ( -.  ph  ->  if ( ph ,  A ,  B )  =  B )
 
Theoremifnefalse 3533 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 3532 directly in this case. It happens, e.g., in oevn0 6468. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  =/=  B  ->  if ( A  =  B ,  C ,  D )  =  D )
 
Theoremifsb 3534 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
 |-  ( if ( ph ,  A ,  B )  =  A  ->  C  =  D )   &    |-  ( if ( ph ,  A ,  B )  =  B  ->  C  =  E )   =>    |-  C  =  if ( ph ,  D ,  E )
 
Theoremdfif3 3535* Alternate definition of the conditional operator df-if 3526. 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 ) ) )
 
Theoremdfif4 3536* Alternate definition of the conditional operator df-if 3526. Note that  ph is independent of  x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  u.  B )  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C ) ) )
 
Theoremdfif5 3537* Alternate definition of the conditional operator df-if 3526. Note that  ph is independent of  x i.e. a constant true or false (see also abvor0 3433). (Contributed by Gérard Lang, 18-Aug-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  i^i  B )  u.  ( ( ( A  \  B )  i^i  C )  u.  ( ( B  \  A )  i^i  ( _V  \  C ) ) ) )
 
Theoremifeq12 3538 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D ) )
 
Theoremifeq1d 3539 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2d 3540 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifeq12d 3541 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 3542 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
 |-  ( ( ph  <->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
 
Theoremifbid 3543 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B ) )
 
Theoremifbieq2i 3544 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 3545 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 3546 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 3547 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 3548 Deduction version of nfif 3549. (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 3549 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 )
 
Theoremifeq1da 3550 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2da 3551 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  -.  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifclda 3552 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  ps )  ->  A  e.  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  e.  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
 
Theoremcsbifg 3553 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by Mario Carneiro, 14-Nov-2016.)
 |-  ( A  e.  V  -> 
 [_ A  /  x ]_ if ( ph ,  B ,  C )  =  if ( [. A  /  x ]. ph ,  [_ A  /  x ]_ B ,  [_ A  /  x ]_ C ) )
 
Theoremelimif 3554 Elimination of a conditional operator contained in a wff  ps. (Contributed by NM, 15-Feb-2005.)
 |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps 
 <->  ch ) )   &    |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th ) )   =>    |-  ( ps  <->  ( ( ph  /\ 
 ch )  \/  ( -.  ph  /\  th )
 ) )
 
Theoremifbothda 3555 A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   &    |-  (
 ( et  /\  ph )  ->  ps )   &    |-  ( ( et 
 /\  -.  ph )  ->  ch )   =>    |-  ( et  ->  th )
 
Theoremifboth 3556 A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   =>    |-  ( ( ps 
 /\  ch )  ->  th )
 
Theoremifid 3557 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
 |- 
 if ( ph ,  A ,  A )  =  A
 
Theoremeqif 3558 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
 |-  ( A  =  if ( ph ,  B ,  C )  <->  ( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) ) )
 
Theoremelif 3559 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
 |-  ( A  e.  if ( ph ,  B ,  C )  <->  ( ( ph  /\  A  e.  B )  \/  ( -.  ph  /\  A  e.  C ) ) )
 
Theoremifel 3560 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
 |-  ( if ( ph ,  A ,  B )  e.  C  <->  ( ( ph  /\  A  e.  C )  \/  ( -.  ph  /\  B  e.  C ) ) )
 
Theoremifcl 3561 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  C  /\  B  e.  C )  ->  if ( ph ,  A ,  B )  e.  C )
 
Theoremifeqor 3562 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
 
Theoremifnot 3563 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
 |- 
 if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
 
Theoremifan 3564 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |- 
 if ( ( ph  /\ 
 ps ) ,  A ,  B )  =  if ( ph ,  if ( ps ,  A ,  B ) ,  B )
 
Theoremifor 3565 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |- 
 if ( ( ph  \/  ps ) ,  A ,  B )  =  if ( ph ,  A ,  if ( ps ,  A ,  B ) )
 
Theoremdedth 3566 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 3573. If the inference has other hypotheses with class variable  A, these can be kept by assigning keephyp 3579 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpegif/mmdeduction.html. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch ) )   &    |-  ch   =>    |-  ( ph  ->  ps )
 
Theoremdedth2h 3567 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3570 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3566. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ch  <->  th ) )   &    |-  ( B  =  if ( ps ,  B ,  D )  ->  ( th 
 <->  ta ) )   &    |-  ta   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremdedth3h 3568 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3567. (Contributed by NM, 15-May-1999.)
 |-  ( 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 3569 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3567. (Contributed by NM, 16-May-1999.)
 |-  ( 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 3570 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 3567 is simpler to use. See also comments in dedth 3566. (Contributed by NM, 13-Aug-1999.) (Proof 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 3571 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3570. (Contributed by NM, 13-Aug-1999.) (Proof 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 3572 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3570. (Contributed by NM, 21-Apr-2007.) (Proof 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 3573 Eliminate a hypothesis containing class variable  A when it is known for a specific class  B. For more information, see comments in dedth 3566. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ph  <->  ps ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <->  ps ) )   &    |-  ch   =>    |-  ps
 
Theoremelimhyp2v 3574 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( 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 3575 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( 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 3576 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3566). (Contributed by NM, 16-Apr-2005.)
 |-  ( 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 3577 Eliminate a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 15-May-1999.)
 |-  B  e.  C   =>    |-  if ( A  e.  C ,  A ,  B )  e.  C
 
Theoremelimdhyp 3578 Version of elimhyp 3573 where the hypothesis is deduced from the final antecedent. See ghomgrplem 23354 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 3579 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. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   &    |-  ps   &    |-  ch   =>    |-  th
 
Theoremkeephyp2v 3580 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3566). (Contributed by NM, 16-Apr-2005.)
 |-  ( 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 3581 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
 |-  ( 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 3582 Keep a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 14-Aug-1999.)
 |-  A  e.  C   &    |-  B  e.  C   =>    |- 
 if ( ph ,  A ,  B )  e.  C
 
Theoremifex 3583 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 if ( ph ,  A ,  B )  e.  _V
 
Theoremifexg 3584 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )
 
2.1.16  Power classes
 
Syntaxcpw 3585 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 3586* Soundness justification theorem for df-pw 3587. (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 3587* 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  =  { 3 ,  5 ,  7 }, then  ~P A  =  { (/) ,  { 3 } ,  { 5 } ,  { 7 } ,  { 3 ,  5 } ,  { 3 ,  7 } ,  {
5 ,  7 } ,  { 3 ,  5 ,  7 } } (ex-pw 20745). We will later introduce the Axiom of Power Sets ax-pow 4146, which can be expressed in class notation per pwexg 4152. Still later we will prove, in hashpw 11339, 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 3588 Equality theorem for power class. (Contributed by NM, 5-Aug-1993.)
 |-  ( A  =  B  ->  ~P A  =  ~P B )
 
Theorempweqi 3589 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
 |-  A  =  B   =>    |-  ~P A  =  ~P B
 
Theorempweqd 3590 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ~P A  =  ~P B )
 
Theoremelpw 3591 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 )
 
Theoremelpwg 3592 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4127. (Contributed by NM, 6-Aug-2000.)
 |-  ( A  e.  V  ->  ( A  e.  ~P B 
 <->  A  C_  B )
 )
 
Theoremelpwi 3593 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  ~P B  ->  A  C_  B )
 
Theoremelpwid 3594 An element of a power class is a subclass. Deduction form of elpwi 3593. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ~P B )   =>    |-  ( ph  ->  A 
 C_  B )
 
Theoremelelpwi 3595 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 3596 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 3597 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  ~P A )
 
Theorempwid 3598 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  A  e.  ~P A
 
Theorempwss 3599* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
 |-  ( ~P A  C_  B 
 <-> 
 A. x ( x 
 C_  A  ->  x  e.  B ) )
 
2.1.17  Unordered and ordered pairs
 
Syntaxcsn 3600 Extend class notation to include singleton.
 class  { A }
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31305
  Copyright terms: Public domain < Previous  Next >