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Theorem iotanul 5234
Description: Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotanul  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )

Proof of Theorem iotanul
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2147 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 5220 . . 3  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1530 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  <->  -.  E. z A. x
( ph  <->  x  =  z
) )
4 ax-1 5 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  ->  -.  A. x
( ph  <->  x  =  z
) ) )
5 eqidd 2284 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
z  =  z )
64, 5impbid1 194 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  <->  -.  A. x ( ph  <->  x  =  z ) ) )
76con2bid 319 . . . . . . . . 9  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( A. x (
ph 
<->  x  =  z )  <->  -.  z  =  z
) )
87alimi 1546 . . . . . . . 8  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  A. z ( A. x ( ph  <->  x  =  z )  <->  -.  z  =  z ) )
9 abbi 2393 . . . . . . . 8  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  -.  z  =  z )  <->  { z  |  A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
108, 9sylib 188 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
11 dfnul2 3457 . . . . . . 7  |-  (/)  =  {
z  |  -.  z  =  z }
1210, 11syl6eqr 2333 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  (/) )
133, 12sylbir 204 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) }  =  (/) )
1413unieqd 3838 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  U. (/) )
15 uni0 3854 . . . 4  |-  U. (/)  =  (/)
1614, 15syl6eq 2331 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  (/) )
172, 16syl5eq 2327 . 2  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  =  (/) )
181, 17sylnbi 297 1  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623   E!weu 2143   {cab 2269   (/)c0 3455   U.cuni 3827   iotacio 5217
This theorem is referenced by:  iotassuni  5235  iotaex  5236  dfiota4  5247  tz6.12-2  5516  dffv3  5521  riotav  6309  riotaprc  6342  isf32lem9  7987  grpidval  14384  0g0  14386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-uni 3828  df-iota 5219
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