Theorem iotanul 5195

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 2029	. . 3 |- ( E! x ph <-> E. z A. x ( ph <-> x = z ) )
2		dfiota2 5181	. . . 4 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
3		alnex 1499	. . . . . . 7 |- ( A. z -. A. x ( ph <-> x = z ) <-> -. E. z A. x ( ph <-> x = z ) )
4		ax-in2 615	. . . . . . . . . 10 |- ( -. A. x ( ph <-> x = z ) -> ( A. x ( ph <-> x = z ) -> -. z = z ) )
5	4	alimi 1455	. . . . . . . . 9 |- ( A. z -. A. x ( ph <-> x = z ) -> A. z ( A. x ( ph <-> x = z ) -> -. z = z ) )
6		ss2ab 3225	. . . . . . . . 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 3426	. . . . . . . 8 |- (/) = { z | -. z = z }
9	7, 8	sseqtrrdi 3206	. . . . . . 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 3835	. . . . 5 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ U. (/) )
12		uni0 3838	. . . . 5 |- U. (/) = (/)
13	11, 12	sseqtrdi 3205	. . . 4 |- ( -. E. z A. x ( ph <-> x = z ) -> U. { z | A. x ( ph <-> x = z ) } C_ (/) )
14	2, 13	eqsstrid 3203	. . 3 |- ( -. E. z A. x ( ph <-> x = z ) -> ( iota x ph ) C_ (/) )
15	1, 14	sylnbi 678	. 2 |- ( -. E! x ph -> ( iota x ph ) C_ (/) )
16		ss0 3465	. 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 1351   = wceq 1353   E.wex 1492   E!weu 2026   {cab 2163   C_ wss 3131   (/)c0 3424   U.cuni 3811   iotacio 5178

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-uni 3812  df-iota 5180

This theorem is referenced by:  tz6.12-2  5508  0fv  5552  riotaund  5867  0g0  12800