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Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremstdpc4-2 26901 Theorem *11.1 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ph  ->  [ z  /  x ] [ w  /  y ] ph )
 
Theorempm11.11 26902 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ph   =>    |- 
 A. z A. w [ z  /  x ] [ w  /  y ] ph
 
Theorempm11.12 26903* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  ->  ( ph  \/  A. x A. y ps )
 )
 
Theorem2exnaln 26904 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ph  <->  -.  A. x A. y  -.  ph )
 
Theorem2nexaln 26905 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph 
 <-> 
 A. x A. y  -.  ph )
 
Theorem19.21vv 26906* Compare theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  ->  ph )  <->  ( ps  ->  A. x A. y ph ) )
 
Theorem2alim 26907 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( A. x A. y ph  ->  A. x A. y ps ) )
 
Theorem2albi 26908 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( A. x A. y ph  <->  A. x A. y ps )
 )
 
Theorem2exim 26909 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( E. x E. y ph  ->  E. x E. y ps ) )
 
Theorem2exbi 26910 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps )
 )
 
Theorema4sbce-2 26911 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  ->  E. x E. y ph )
 
Theorem19.33-2 26912 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x A. y ph  \/  A. x A. y ps )  ->  A. x A. y ( ph  \/  ps ) )
 
Theorem19.36vv 26913* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y (
 ph  ->  ps )  <->  ( A. x A. y ph  ->  ps )
 )
 
Theorem19.31vv 26914* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  <->  (
 A. x A. y ph  \/  ps ) )
 
Theorem19.37vv 26915* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
 
Theorem19.28vv 26916* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  /\  ph )  <->  ( ps  /\  A. x A. y ph ) )
 
Theorempm11.52 26917 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  -. 
 A. x A. y
 ( ph  ->  -.  ps ) )
 
Theorem2exanali 26918 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y
 ( ph  /\  -.  ps ) 
 <-> 
 A. x A. y
 ( ph  ->  ps )
 )
 
Theoremaaanv 26919* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 1811. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ph  /\  A. y ps )  <->  A. x A. y
 ( ph  /\  ps )
 )
 
Theorempm11.57 26920* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ph  <->  A. x A. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.58 26921* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  <->  E. x E. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.59 26922* Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( A. x ( ph  ->  ps )  ->  A. y A. x ( ( ph  /\ 
 [ y  /  x ] ph )  ->  ( ps  /\  [ y  /  x ] ps ) ) )
 
Theorempm11.6 26923* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x ( E. y
 ( ph  /\  ps )  /\  ch )  <->  E. y ( E. x ( ph  /\  ch )  /\  ps ) )
 
Theorempm11.61 26924* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. y A. x (
 ph  ->  ps )  ->  A. x ( ph  ->  E. y ps ) )
 
Theorempm11.62 26925* Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ( ph  /\  ps )  ->  ch )  <->  A. x ( ph  ->  A. y ( ps 
 ->  ch ) ) )
 
Theorempm11.63 26926 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph  ->  A. x A. y
 ( ph  ->  ps )
 )
 
Theorempm11.7 26927 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  \/  ph )  <->  E. x E. y ph )
 
Theorempm11.71 26928* Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x ph  /\ 
 E. y ch )  ->  ( ( A. x ( ph  ->  ps )  /\  A. y ( ch 
 ->  th ) )  <->  A. x A. y
 ( ( ph  /\  ch )  ->  ( ps  /\  th ) ) ) )
 
16.18.3  Predicate Calculus
 
Theoremsbeqal1 26929* If  x  =  y always implies 
x  =  z, then  y  =  z is true. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
 
Theoremsbeqal1i 26930* Suppose you know  x  =  y implies  x  =  z, assuming  x and  z are distinct. Then,  y  =  z. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  y  =  z
 
Theoremsbeqal2i 26931* If  x  =  y implies  x  =  z, then we can infer  z  =  y. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  z  =  y
 
Theoremsbeqalbi 26932* When both  x and  z and  y and  z are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( x  =  y  <->  A. z ( z  =  x  ->  z  =  y ) )
 
Theoremax4567 26933 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1536 as the inference rule. This proof extends the idea of ax467 1752 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
 |-  (
 ( A. x A. y  -.  A. x A. y
 ( A. y ph  ->  ps )  ->  ( ph  ->  A. y ( A. y ph  ->  ps )
 ) )  ->  ( A. y ph  ->  A. y ps ) )
 
Theoremax4567to4 26934 Re-derivation of ax-4 1692 from ax4567 26933. Note that ax-9 1684 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremax4567to5 26935 Re-derivation of ax-5o 1694 from ax4567 26933. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ( A. x ph 
 ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremax4567to6 26936 Re-derivation of ax-6o 1697 from ax4567 26933. Note that neither ax-6o 1697 nor ax-7 1535 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax4567to7 26937 Re-derivation of ax-7 1535 from ax4567 26933. Note that ax-7 1535 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax10ext 26938* This theorem shows that, given axext4 2240, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
Theoremax10-16 26939* This theorem shows that, given ax-16 1927, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
16.18.4  Principia Mathematica * 13 and * 14
 
Theorempm13.13a 26940 One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  A )  ->  [. A  /  x ]. ph )
 
Theorempm13.13b 26941 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  x  =  A )  ->  ph )
 
Theorempm13.14 26942 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  -.  ph )  ->  x  =/=  A )
 
Theorempm13.192 26943* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( A. x ( x  =  A  <->  x  =  y )  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.193 26944 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  x  =  y ) )
 
Theorempm13.194 26945 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y ) )
 
Theorempm13.195 26946* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 2959. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( y  =  A  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.196a 26947* Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( -.  ph  <->  A. y ( [
 y  /  x ] ph  ->  y  =/=  x ) )
 
Theorem2sbc6g 26948* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theorem2sbc5g 26949* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theoremiotain 26950 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( E! x ph  ->  |^| { x  |  ph }  =  (
 iota x ph ) )
 
Theoremiotaexeu 26951 The iota class exists. This theorem does not require ax-nul 4089 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e. 
 _V )
 
Theoremiotasbc 26952* Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define  iota in terms of a function of  ( iota x ph ). Their definition differs in that a function of  ( iota x ph ) evaluates to "false" when there isn't a single  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x ( ph  <->  x  =  y
 )  /\  ps )
 ) )
 
Theoremiotasbc2 26953* Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  (
 ( E! x ph  /\ 
 E! x ps )  ->  ( [. ( iota
 x ph )  /  y ]. [. ( iota x ps )  /  z ]. ch  <->  E. y E. z
 ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z
 )  /\  ch )
 ) )
 
Theorempm14.12 26954* Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  A. x A. y ( ( ph  /\  [. y  /  x ].
 ph )  ->  x  =  y ) )
 
Theorempm14.122a 26955* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph )
 ) )
 
Theorempm14.122b 26956* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  ( ( A. x (
 ph  ->  x  =  A )  /\  [. A  /  x ].
 ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.122c 26957* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.123a 26958* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph ) ) )
 
Theorempm14.123b 26959* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.123c 26960* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.18 26961 Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ps  ->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremiotaequ 26962* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x x  =  y )  =  y
 
Theoremiotavalb 26963* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6201. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y
 ) 
 <->  ( iota x ph )  =  y )
 )
 
Theoremiotasbc5 26964* Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) ) )
 
Theorempm14.24 26965* Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  A. y
 ( [. y  /  x ].
 ph 
 <->  y  =  ( iota
 x ph ) ) )
 
Theoremiotavalsb 26966* Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  x  =  y
 )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps )
 )
 
Theoremsbiota1 26967 Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremsbaniota 26968 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( E. x ( ph  /\  ps ) 
 <-> 
 [. ( iota x ph )  /  x ]. ps ) )
 
Theoremeubi 26969 Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  ps )  ->  ( E! x ph  <->  E! x ps )
 )
 
Theoremiotasbcq 26970 Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  ps )  ->  ( [. ( iota x ph )  /  y ]. ch  <->  [. ( iota x ps )  /  y ]. ch )
 )
 
16.18.5  Set Theory
 
Theoremelnev 26971* Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)
 |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
 
TheoremrusbcALT 26972 A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (Proof modification is discouraged.)
 |-  { x  |  x  e/  x }  e/  _V
 
Theoremcompel 26973 Equivalence between two ways of saying "is a member of the complement of  A." (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
 
Theoremcompeq 26974* Equality between two ways of saying "the complement of  A." (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( _V  \  A )  =  { x  |  -.  x  e.  A }
 
Theoremcompne 26975 The complement of  A is not equal to  A. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( _V  \  A )  =/= 
 A
 
TheoremcompneOLD 26976 Obsolete proof of compne 26975 as of 28-Jun-2015. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( _V  \  A )  =/= 
 A
 
Theoremcompab 26977 Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( _V  \  { z  | 
 ph } )  =  { z  |  -.  ph
 }
 
Theoremconss34 26978 Contrpositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( A  C_  B  <->  ( _V  \  B )  C_  ( _V  \  A ) )
 
Theoremconss2 26979 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( A  C_  ( _V  \  B ) 
 <->  B  C_  ( _V  \  A ) )
 
Theoremconss1 26980 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  (
 ( _V  \  A )  C_  B  <->  ( _V  \  B )  C_  A )
 
Theoremralbidar 26981 More general form of ralbida 2528. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidar 26982 More general form of rexbida 2529. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremdropab1 26983 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. x ,  z >.  |  ph }  =  { <. y ,  z >.  |  ph } )
 
Theoremdropab2 26984 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. z ,  x >.  |  ph }  =  { <. z ,  y >.  |  ph } )
 
Theoremipo0 26985 If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Po  A  <->  A  =  (/) )
 
Theoremifr0 26986 A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Fr  A  <->  A  =  (/) )
 
Theoremordpss 26987 ordelpss 4357 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( Ord  B  ->  ( A  e.  B  ->  A  C.  B ) )
 
Theoremfvsb 26988* Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
Theoremfveqsb 26989* Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  ( F `  A )  ->  ( ph 
 <->  ps ) )   &    |-  F/ x ps   =>    |-  ( E! y  A F y  ->  ( ps 
 <-> 
 E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
TheoremxrltneNEW 26990 'Less than' implies not equal for extended reals. (Contributed by Andrew Salmon, 11-Nov-2011.)
 |-  (
 ( A  e.  RR*  /\  A  <  B ) 
 ->  A  =/=  B )
 
Theoremxpexb 26991 A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
 
Theoremxpexcnv 26992 A condition where the converse of xpex 4754 holds as well. Corollary 6.9(2) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( B  =/=  (/)  /\  ( A  X.  B )  e. 
 _V )  ->  A  e.  _V )
 
Theoremtrelpss 26993 An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4314, ax-reg 7239 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( Tr  A  /\  B  e.  A )  ->  B  C.  A )
 
16.18.6  Arithmetic
 
Theoremaddcomgi 26994 Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( A  +  B )  =  ( B  +  A )
 
16.18.7  Geometry
 
Syntaxcplusr 26995 Introduce the operation of vector addition.
 class  + r
 
Syntaxcminusr 26996 Introduce the operation of vector subtraction.
 class  - r
 
Syntaxctimesr 26997 Introduce the operation of scalar multiplication.
 class  . v
 
Syntaxcptdfc 26998  PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.
 class  PtDf ( A ,  B )
 
Syntaxcrr3c 26999  RR 3 is a class.
 class  RR 3
 
Syntaxcline3 27000  line 3 is a class.
 class  line 3
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