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Theorem iotanul 5396
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 2257 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 5382 . . 3  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1588 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  <->  -.  E. z A. x
( ph  <->  x  =  z
) )
4 ax-1 6 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  ->  -.  A. x
( ph  <->  x  =  z
) ) )
5 eqidd 2444 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
z  =  z )
64, 5impbid1 203 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  <->  -.  A. x ( ph  <->  x  =  z ) ) )
76con2bid 329 . . . . . . . . 9  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( A. x (
ph 
<->  x  =  z )  <->  -.  z  =  z
) )
87alimi 1604 . . . . . . . 8  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  A. z ( A. x ( ph  <->  x  =  z )  <->  -.  z  =  z ) )
9 abbi 2553 . . . . . . . 8  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  -.  z  =  z )  <->  { z  |  A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
108, 9sylib 196 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
11 dfnul2 3639 . . . . . . 7  |-  (/)  =  {
z  |  -.  z  =  z }
1210, 11syl6eqr 2493 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  (/) )
133, 12sylbir 213 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) }  =  (/) )
1413unieqd 4101 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  U. (/) )
15 uni0 4118 . . . 4  |-  U. (/)  =  (/)
1614, 15syl6eq 2491 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  (/) )
172, 16syl5eq 2487 . 2  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  =  (/) )
181, 17sylnbi 306 1  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1367    = wceq 1369   E.wex 1586   E!weu 2253   {cab 2429   (/)c0 3637   U.cuni 4091   iotacio 5379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-v 2974  df-dif 3331  df-in 3335  df-ss 3342  df-nul 3638  df-sn 3878  df-uni 4092  df-iota 5381
This theorem is referenced by:  iotassuni  5397  iotaex  5398  dfiota4  5409  csbiota  5410  tz6.12-2  5682  dffv3  5687  csbriota  6064  riotaund  6088  isf32lem9  8530  grpidval  15432  0g0  15434
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