Theorem iotanul 5175

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 2022	. . 3 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		dfiota2 5161	. . . 4 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
3		alnex 1492	. . . . . . 7 |- ( A. z -. A. x ( ph <-> x = z ) <-> -. E. z A. x ( ph <-> x = z ) )
4		ax-in2 610	. . . . . . . . . 10 |- ( -. A. x ( ph <-> x = z ) -> ( A. x ( ph <-> x = z ) -> -. z = z ) )
5	4	alimi 1448	. . . . . . . . 9 |- ( A. z -. A. x ( ph <-> x = z ) -> A. z ( A. x ( ph <-> x = z ) -> -. z = z ) )
6		ss2ab 3215	. . . . . . . . 9 |- ( { z | A. x ( ph <-> x = z ) } C_ { z | -. z = z } <-> A. z ( A. x ( ph <-> x = z ) -> -. z = z ) )
7	5, 6	sylibr 133	. . . . . . . 8 |- ( A. z -. A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } C_ { z | -. z = z } )
8		dfnul2 3416	. . . . . . . 8 |- (/) = { z | -. z = z }
9	7, 8	sseqtrrdi 3196	. . . . . . 7 |- ( A. z -. A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } C_ (/) )
10	3, 9	sylbir 134	. . . . . 6 |- ( -. E. z A. x ( ph <-> x = z ) -> { z | A. x ( ph <-> x = z ) } C_ (/) )
11	10	unissd 3820	. . . . 5 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ U. (/) )
12		uni0 3823	. . . . 5 |- U. (/) = (/)
13	11, 12	sseqtrdi 3195	. . . 4 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ (/) )
14	2, 13	eqsstrid 3193	. . 3 |- ( -. E. z A. x ( ph <-> x = z ) -> ( iota x ph ) C_ (/) )
15	1, 14	sylnbi 673	. 2 |- ( -. E! x ph -> ( iota x ph ) C_ (/) )
16		ss0 3455	. 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 104   A.wal 1346   = wceq 1348   E.wex 1485   E!weu 2019   {cab 2156   C_ wss 3121   (/)c0 3414   U.cuni 3796   iotacio 5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-uni 3797  df-iota 5160

This theorem is referenced by:  tz6.12-2  5487  0fv  5531  riotaund  5843  0g0  12630