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Theorem iotaex 5426
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 5420 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
21eqcomd 2440 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
32eximi 1585 . . 3  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  E. z  z  =  ( iota x ph ) )
4 df-eu 2284 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
5 isset 2952 . . 3  |-  ( ( iota x ph )  e.  _V  <->  E. z  z  =  ( iota x ph ) )
63, 4, 53imtr4i 258 . 2  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
7 iotanul 5424 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
8 0ex 4331 . . 3  |-  (/)  e.  _V
97, 8syl6eqel 2523 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  e.  _V )
106, 9pm2.61i 158 1  |-  ( iota
x ph )  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   _Vcvv 2948   (/)c0 3620   iotacio 5407
This theorem is referenced by:  iota4an  5428  fvex  5733  riotaex  6544  erov  6992  iunfictbso  7984  isf32lem9  8230  sumex  12469  pcval  13206  grpidval  14695  fn0g  14696  gsumvalx  14762  dchrptlem1  21036  lgsdchrval  21119  lgsdchr  21120  prodex  25222  psgnfn  27339  psgnval  27345  bnj1366  29055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-uni 4008  df-iota 5409
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