Theorem iotanul 5293

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 2080	. . 3 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		dfiota2 5278	. . . 4 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
3		alnex 1545	. . . . . . 7 |- ( A. z -. A. x ( ph <-> x = z ) <-> -. E. z A. x ( ph <-> x = z ) )
4		ax-in2 618	. . . . . . . . . 10 |- ( -. A. x ( ph <-> x = z ) -> ( A. x ( ph <-> x = z ) -> -. z = z ) )
5	4	alimi 1501	. . . . . . . . 9 |- ( A. z -. A. x ( ph <-> x = z ) -> A. z ( A. x ( ph <-> x = z ) -> -. z = z ) )
6		ss2ab 3292	. . . . . . . . 9 |- ( { z | A. x ( ph <-> x = z ) } C_ { z | -. z = z } <-> A. z ( A. x ( ph <-> x = z ) -> -. z = z ) )
7	5, 6	sylibr 134	. . . . . . . 8 |- ( A. z -. A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } C_ { z | -. z = z } )
8		dfnul2 3493	. . . . . . . 8 |- (/) = { z | -. z = z }
9	7, 8	sseqtrrdi 3273	. . . . . . 7 |- ( A. z -. A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } C_ (/) )
10	3, 9	sylbir 135	. . . . . 6 |- ( -. E. z A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } C_ (/) )
11	10	unissd 3911	. . . . 5 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ U. (/) )
12		uni0 3914	. . . . 5 |- U. (/) = (/)
13	11, 12	sseqtrdi 3272	. . . 4 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ (/) )
14	2, 13	eqsstrid 3270	. . 3 |- ( -. E. z A. x ( ph <-> x = z ) -> ( iota x ph ) C_ (/) )
15	1, 14	sylnbi 682	. 2 |- ( -. E! x ph -> ( iota x ph ) C_ (/) )
16		ss0 3532	. 2 |- ( ( iota x ph ) C_ (/) -> ( iota x ph ) = (/) )
17	15, 16	syl 14	1 |- ( -. E! x ph -> ( iota x ph ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   E!weu 2077   {cab 2215   C_ wss 3197   (/)c0 3491   U.cuni 3887   iotacio 5275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-uni 3888  df-iota 5277

This theorem is referenced by:  tz6.12-2  5617  0fv  5664  riotaund  5990  0g0  13404