MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iotanul Unicode version

Theorem iotanul 5424
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 2284 . 2  |-  ( E! x ph  <->  E. z A. x ( ph  <->  x  =  z ) )
2 dfiota2 5410 . . 3  |-  ( iota
x ph )  =  U. { z  |  A. x ( ph  <->  x  =  z ) }
3 alnex 1552 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  <->  -.  E. z A. x
( ph  <->  x  =  z
) )
4 ax-1 5 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  ->  -.  A. x
( ph  <->  x  =  z
) ) )
5 eqidd 2436 . . . . . . . . . . 11  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
z  =  z )
64, 5impbid1 195 . . . . . . . . . 10  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( z  =  z  <->  -.  A. x ( ph  <->  x  =  z ) ) )
76con2bid 320 . . . . . . . . 9  |-  ( -. 
A. x ( ph  <->  x  =  z )  -> 
( A. x (
ph 
<->  x  =  z )  <->  -.  z  =  z
) )
87alimi 1568 . . . . . . . 8  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  A. z ( A. x ( ph  <->  x  =  z )  <->  -.  z  =  z ) )
9 abbi 2545 . . . . . . . 8  |-  ( A. z ( A. x
( ph  <->  x  =  z
)  <->  -.  z  =  z )  <->  { z  |  A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
108, 9sylib 189 . . . . . . 7  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  { z  |  -.  z  =  z } )
11 dfnul2 3622 . . . . . . 7  |-  (/)  =  {
z  |  -.  z  =  z }
1210, 11syl6eqr 2485 . . . . . 6  |-  ( A. z  -.  A. x (
ph 
<->  x  =  z )  ->  { z  | 
A. x ( ph  <->  x  =  z ) }  =  (/) )
133, 12sylbir 205 . . . . 5  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  { z  |  A. x ( ph  <->  x  =  z ) }  =  (/) )
1413unieqd 4018 . . . 4  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  U. (/) )
15 uni0 4034 . . . 4  |-  U. (/)  =  (/)
1614, 15syl6eq 2483 . . 3  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  U. { z  |  A. x (
ph 
<->  x  =  z ) }  =  (/) )
172, 16syl5eq 2479 . 2  |-  ( -. 
E. z A. x
( ph  <->  x  =  z
)  ->  ( iota x ph )  =  (/) )
181, 17sylnbi 298 1  |-  ( -.  E! x ph  ->  ( iota x ph )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652   E!weu 2280   {cab 2421   (/)c0 3620   U.cuni 4007   iotacio 5407
This theorem is referenced by:  iotassuni  5425  iotaex  5426  dfiota4  5437  tz6.12-2  5710  dffv3  5715  riotav  6545  riotaprc  6578  isf32lem9  8230  grpidval  14695  0g0  14697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-uni 4008  df-iota 5409
  Copyright terms: Public domain W3C validator