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Theorem iotanul 5568
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 2323 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 5554 . . 3  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1673 . . . . . 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 2472 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
z  =  z )
64, 5impbid1 208 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  <->  -.  A. x ( ph  <->  x  =  z ) ) )
76con2bid 336 . . . . . . . . 9  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( A. x (
ph 
<->  x  =  z )  <->  -.  z  =  z
) )
87alimi 1692 . . . . . . . 8  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  A. z ( A. x ( ph  <->  x  =  z )  <->  -.  z  =  z ) )
9 abbi 2585 . . . . . . . 8  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  -.  z  =  z )  <->  { z  |  A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
108, 9sylib 201 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
11 dfnul2 3724 . . . . . . 7  |-  (/)  =  {
z  |  -.  z  =  z }
1210, 11syl6eqr 2523 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  (/) )
133, 12sylbir 218 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) }  =  (/) )
1413unieqd 4200 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  U. (/) )
15 uni0 4217 . . . 4  |-  U. (/)  =  (/)
1614, 15syl6eq 2521 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  (/) )
172, 16syl5eq 2517 . 2  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  =  (/) )
181, 17sylnbi 313 1  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189   A.wal 1450    = wceq 1452   E.wex 1671   E!weu 2319   {cab 2457   (/)c0 3722   U.cuni 4190   iotacio 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-sn 3960  df-uni 4191  df-iota 5553
This theorem is referenced by:  iotassuni  5569  iotaex  5570  dfiota4  5581  csbiota  5582  tz6.12-2  5870  dffv3  5875  csbriota  6282  riotaund  6305  isf32lem9  8809  grpidval  16581  0g0  16584
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