Theorem iotanul 5866

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.

Step	Hyp	Ref	Expression
1		df-eu 2474	. 2 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		dfiota2 5852	. . 3 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
3		alnex 1706	. . . . . 6 |- ( A. z -. A. x ( ph <-> x = z ) <-> -. E. z A. x ( ph <-> x = z ) )
4		dfnul2 3917	. . . . . . 7 |- (/) = { z | -. z = z }
5		equid 1939	. . . . . . . . . . . 12 |- z = z
6	5	tbt 359	. . . . . . . . . . 11 |- ( -. A. x ( ph <-> x = z ) <-> ( -. A. x ( ph <-> x = z ) <-> z = z ) )
7	6	biimpi 206	. . . . . . . . . 10 |- ( -. A. x ( ph <-> x = z ) -> ( -. A. x ( ph <-> x = z ) <-> z = z ) )
8	7	con1bid 345	. . . . . . . . 9 |- ( -. A. x ( ph <-> x = z ) -> ( -. z = z <-> A. x ( ph <-> x = z ) ) )
9	8	alimi 1739	. . . . . . . 8 |- ( A. z -. A. x ( ph <-> x = z ) -> A. z ( -. z = z <-> A. x ( ph <-> x = z ) ) )
10		abbi 2737	. . . . . . . 8 |- ( A. z ( -. z = z <-> A. x ( ph <-> x = z ) ) <-> { z | -. z = z } = { z | A. x ( ph <-> x = z ) } )
11	9, 10	sylib 208	. . . . . . 7 |- ( A. z -. A. x ( ph <-> x = z ) -> { z | -. z = z } = { z | A. x ( ph <-> x = z ) } )
12	4, 11	syl5req 2669	. . . . . 6 |- ( A. z -. A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } = (/) )
13	3, 12	sylbir 225	. . . . 5 |- ( -. E. z A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } = (/) )
14	13	unieqd 4446	. . . 4 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } = U. (/) )
15		uni0 4465	. . . 4 |- U. (/) = (/)
16	14, 15	syl6eq 2672	. . 3 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } = (/) )
17	2, 16	syl5eq 2668	. 2 |- ( -. E. z A. x ( ph <-> x = z ) -> ( iota x ph ) = (/) )
18	1, 17	sylnbi 320	1 |- ( -. E! x ph -> ( iota x ph ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   E!weu 2470   {cab 2608   (/)c0 3915   U.cuni 4436   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437  df-iota 5851

This theorem is referenced by:  iotassuni  5867  iotaex  5868  dfiota4  5879  dfiota4OLD  5880  csbiota  5881  tz6.12-2  6182  dffv3  6187  csbriota  6623  riotaund  6647  isf32lem9  9183  grpidval  17260  0g0  17263