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Theorem iotanul 4961
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 1948 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 4947 . . . 4  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1431 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  <->  -.  E. z A. x
( ph  <->  x  =  z
) )
4 ax-in2 578 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( A. x (
ph 
<->  x  =  z )  ->  -.  z  =  z ) )
54alimi 1387 . . . . . . . . 9  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  A. z ( A. x ( ph  <->  x  =  z )  ->  -.  z  =  z )
)
6 ss2ab 3078 . . . . . . . . 9  |-  ( { z  |  A. x
( ph  <->  x  =  z
) }  C_  { z  |  -.  z  =  z }  <->  A. z
( A. x (
ph 
<->  x  =  z )  ->  -.  z  =  z ) )
75, 6sylibr 132 . . . . . . . 8  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) } 
C_  { z  |  -.  z  =  z } )
8 dfnul2 3277 . . . . . . . 8  |-  (/)  =  {
z  |  -.  z  =  z }
97, 8syl6sseqr 3062 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) } 
C_  (/) )
103, 9sylbir 133 . . . . . 6  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) } 
C_  (/) )
1110unissd 3660 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  C_  U. (/) )
12 uni0 3663 . . . . 5  |-  U. (/)  =  (/)
1311, 12syl6sseq 3061 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  C_  (/) )
142, 13syl5eqss 3059 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  C_  (/) )
151, 14sylnbi 636 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  C_  (/) )
16 ss0 3311 . 2  |-  ( ( iota x ph )  C_  (/)  ->  ( iota x ph )  =  (/) )
1715, 16syl 14 1  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103   A.wal 1285    = wceq 1287   E.wex 1424   E!weu 1945   {cab 2071    C_ wss 2988   (/)c0 3275   U.cuni 3636   iotacio 4944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-uni 3637  df-iota 4946
This theorem is referenced by:  tz6.12-2  5259  0fv  5302  riotaund  5603
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