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Theorem iotaex 6160
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
StepHypRef Expression
1 iotaval 6154 . . . . 5  |-  ( A. x ( ph  <->  x  =  z )  ->  ( iota x ph )  =  z )
21eqcomd 2258 . . . 4  |-  ( A. x ( ph  <->  x  =  z )  ->  z  =  ( iota x ph ) )
32eximi 1574 . . 3  |-  ( E. z A. x (
ph 
<->  x  =  z )  ->  E. z  z  =  ( iota x ph ) )
4 df-eu 2118 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
5 isset 2731 . . 3  |-  ( ( iota x ph )  e.  _V  <->  E. z  z  =  ( iota x ph ) )
63, 4, 53imtr4i 259 . 2  |-  ( E! x ph  ->  ( iota x ph )  e. 
_V )
7 iotanul 6158 . . 3  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
8 0ex 4047 . . 3  |-  (/)  e.  _V
97, 8syl6eqel 2341 . 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 5    <-> wb 178   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   E!weu 2114   _Vcvv 2727   (/)c0 3362   iotacio 6141
This theorem is referenced by:  iota4an  6162  riotaex  6194  erov  6641  iunfictbso  7625  isf32lem9  7871  sumex  12037  pcval  12771  grpidval  14219  fn0g  14220  gsumvalx  14286  dchrptlem1  20335  lgsdchrval  20418  lgsdchr  20419  psgnfn  26590  psgnval  26596  bnj1366  27551
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-nul 4046
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-sn 3550  df-pr 3551  df-uni 3728  df-iota 6143
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