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Theorem iotanul 6240
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
StepHypRef Expression
1 df-eu 2122 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 6226 . . 3  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1569 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  <->  -.  E. z A. x
( ph  <->  x  =  z
) )
4 ax-1 7 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  ->  -.  A. x
( ph  <->  x  =  z
) ) )
5 eqidd 2259 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
z  =  z )
64, 5impbid1 196 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  <->  -.  A. x ( ph  <->  x  =  z ) ) )
76con2bid 321 . . . . . . . . 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 2368 . . . . . . . 8  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  -.  z  =  z )  <->  { z  |  A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
108, 9sylib 190 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
11 dfnul2 3432 . . . . . . 7  |-  (/)  =  {
z  |  -.  z  =  z }
1210, 11syl6eqr 2308 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  (/) )
133, 12sylbir 206 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) }  =  (/) )
1413unieqd 3812 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  U. (/) )
15 uni0 3828 . . . 4  |-  U. (/)  =  (/)
1614, 15syl6eq 2306 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  (/) )
172, 16syl5eq 2302 . 2  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  =  (/) )
181, 17sylnbi 299 1  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178   A.wal 1532   E.wex 1537    = wceq 1619   E!weu 2118   {cab 2244   (/)c0 3430   U.cuni 3801   iotacio 6223
This theorem is referenced by:  iotassuni  6241  iotaex  6242  riotav  6277  riotaprc  6310  isf32lem9  7955  grpidval  14346  0g0  14348
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 1927  ax-ext 2239
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 1884  df-eu 2122  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-v 2765  df-dif 3130  df-in 3134  df-ss 3141  df-nul 3431  df-sn 3620  df-uni 3802  df-iota 6225
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