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Theorem iotaex 5238
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 5232 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
21eqcomd 2290 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
32eximi 1565 . . 3  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  E. z  z  =  ( iota x ph ) )
4 df-eu 2149 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
5 isset 2794 . . 3  |-  ( ( iota x ph )  e.  _V  <->  E. z  z  =  ( iota x ph ) )
63, 4, 53imtr4i 257 . 2  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
7 iotanul 5236 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
8 0ex 4152 . . 3  |-  (/)  e.  _V
97, 8syl6eqel 2373 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  e.  _V )
106, 9pm2.61i 156 1  |-  ( iota
x ph )  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   A.wal 1529   E.wex 1530    = wceq 1625    e. wcel 1686   E!weu 2145   _Vcvv 2790   (/)c0 3457   iotacio 5219
This theorem is referenced by:  iota4an  5240  fvex  5541  riotaex  6310  erov  6757  iunfictbso  7743  isf32lem9  7989  sumex  12162  pcval  12899  grpidval  14386  fn0g  14387  gsumvalx  14453  dchrptlem1  20505  lgsdchrval  20588  lgsdchr  20589  psgnfn  27435  psgnval  27441  bnj1366  28935
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-nul 4151
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-sn 3648  df-pr 3649  df-uni 3830  df-iota 5221
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