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Theorem iotanul 4882
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 1903 . . 3  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 4868 . . . 4  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1388 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  <->  -.  E. z A. x
( ph  <->  x  =  z
) )
4 ax-in2 545 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( A. x (
ph 
<->  x  =  z )  ->  -.  z  =  z ) )
54alimi 1344 . . . . . . . . 9  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  A. z ( A. x ( ph  <->  x  =  z )  ->  -.  z  =  z )
)
6 ss2ab 3008 . . . . . . . . 9  |-  ( { z  |  A. x
( ph  <->  x  =  z
) }  C_  { z  |  -.  z  =  z }  <->  A. z
( A. x (
ph 
<->  x  =  z )  ->  -.  z  =  z ) )
75, 6sylibr 137 . . . . . . . 8  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) } 
C_  { z  |  -.  z  =  z } )
8 dfnul2 3226 . . . . . . . 8  |-  (/)  =  {
z  |  -.  z  =  z }
97, 8syl6sseqr 2992 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) } 
C_  (/) )
103, 9sylbir 125 . . . . . 6  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) } 
C_  (/) )
1110unissd 3604 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  C_  U. (/) )
12 uni0 3607 . . . . 5  |-  U. (/)  =  (/)
1311, 12syl6sseq 2991 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  C_  (/) )
142, 13syl5eqss 2989 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  C_  (/) )
151, 14sylnbi 603 . 2  |-  ( -.  E! x ph  ->  ( iota x ph )  C_  (/) )
16 ss0 3257 . 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 98   A.wal 1241    = wceq 1243   E.wex 1381   E!weu 1900   {cab 2026    C_ wss 2917   (/)c0 3224   U.cuni 3580   iotacio 4865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-uni 3581  df-iota 4867
This theorem is referenced by:  tz6.12-2  5169  0fv  5208  riotaund  5502
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