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Theorem iotanul 5073
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 1980 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 5059 . . . 4  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1460 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  <->  -.  E. z A. x
( ph  <->  x  =  z
) )
4 ax-in2 589 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( A. x (
ph 
<->  x  =  z )  ->  -.  z  =  z ) )
54alimi 1416 . . . . . . . . 9  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  A. z ( A. x ( ph  <->  x  =  z )  ->  -.  z  =  z )
)
6 ss2ab 3135 . . . . . . . . 9  |-  ( { z  |  A. x
( ph  <->  x  =  z
) }  C_  { z  |  -.  z  =  z }  <->  A. z
( A. x (
ph 
<->  x  =  z )  ->  -.  z  =  z ) )
75, 6sylibr 133 . . . . . . . 8  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) } 
C_  { z  |  -.  z  =  z } )
8 dfnul2 3335 . . . . . . . 8  |-  (/)  =  {
z  |  -.  z  =  z }
97, 8sseqtrrdi 3116 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) } 
C_  (/) )
103, 9sylbir 134 . . . . . 6  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) } 
C_  (/) )
1110unissd 3730 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  C_  U. (/) )
12 uni0 3733 . . . . 5  |-  U. (/)  =  (/)
1311, 12sseqtrdi 3115 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  C_  (/) )
142, 13eqsstrid 3113 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  C_  (/) )
151, 14sylnbi 652 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  C_  (/) )
16 ss0 3373 . 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 104   A.wal 1314    = wceq 1316   E.wex 1453   E!weu 1977   {cab 2103    C_ wss 3041   (/)c0 3333   U.cuni 3706   iotacio 5056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-uni 3707  df-iota 5058
This theorem is referenced by:  tz6.12-2  5380  0fv  5424  riotaund  5732
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