MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iotaex Structured version   Visualization version   Unicode version

Theorem iotaex 5570
Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the  iota class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaex  |-  ( iota
x ph )  e.  _V

Proof of Theorem iotaex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 iotaval 5564 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
21eqcomd 2477 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
32eximi 1715 . . 3  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  E. z  z  =  ( iota x ph ) )
4 df-eu 2323 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
5 isset 3035 . . 3  |-  ( ( iota x ph )  e.  _V  <->  E. z  z  =  ( iota x ph ) )
63, 4, 53imtr4i 274 . 2  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
7 iotanul 5568 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
8 0ex 4528 . . 3  |-  (/)  e.  _V
97, 8syl6eqel 2557 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  e.  _V )
106, 9pm2.61i 169 1  |-  ( iota
x ph )  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E!weu 2319   _Vcvv 3031   (/)c0 3722   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960  df-pr 3962  df-uni 4191  df-iota 5553
This theorem is referenced by:  iota4an  5572  fvex  5889  riotaex  6274  erov  7478  iunfictbso  8563  isf32lem9  8809  sumex  13831  prodex  14038  pcval  14873  grpidval  16581  fn0g  16583  gsumvalx  16591  psgnfn  17220  psgnval  17226  dchrptlem1  24271  lgsdchrval  24354  lgsdchr  24355  bnj1366  29713  bj-finsumval0  31772  ellimciota  37791  fourierdlem36  38118
  Copyright terms: Public domain W3C validator