Theorem iotanul 5309

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 2082	. . 3 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		dfiota2 5294	. . . 4 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
3		alnex 1548	. . . . . . 7 |- ( A. z -. A. x ( ph <-> x = z ) <-> -. E. z A. x ( ph <-> x = z ) )
4		ax-in2 620	. . . . . . . . . 10 |- ( -. A. x ( ph <-> x = z ) -> ( A. x ( ph <-> x = z ) -> -. z = z ) )
5	4	alimi 1504	. . . . . . . . 9 |- ( A. z -. A. x ( ph <-> x = z ) -> A. z ( A. x ( ph <-> x = z ) -> -. z = z ) )
6		ss2ab 3296	. . . . . . . . 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 3498	. . . . . . . 8 |- (/) = { z | -. z = z }
9	7, 8	sseqtrrdi 3277	. . . . . . 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 3922	. . . . 5 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ U. (/) )
12		uni0 3925	. . . . 5 |- U. (/) = (/)
13	11, 12	sseqtrdi 3276	. . . 4 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ (/) )
14	2, 13	eqsstrid 3274	. . 3 |- ( -. E. z A. x ( ph <-> x = z ) -> ( iota x ph ) C_ (/) )
15	1, 14	sylnbi 685	. 2 |- ( -. E! x ph -> ( iota x ph ) C_ (/) )
16		ss0 3537	. 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 1396   = wceq 1398   E.wex 1541   E!weu 2079   {cab 2217   C_ wss 3201   (/)c0 3496   U.cuni 3898   iotacio 5291

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-uni 3899  df-iota 5293

This theorem is referenced by:  tz6.12-2  5639  0fv  5686  riotaund  6018  0g0  13522