Theorem iotanul 4355

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!xph -> (iotaxph) = (/))

Proof of Theorem iotanul

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2208	. 2 |- (E!xph <-> E.zA.x(ph <-> x = z))
2		dfiota2 4341	. . 3 |- (iotaxph) = U.{z | A.x(ph <-> x = z)}
3		alnex 1543	. . . . . 6 |- (A.z -. A.x(ph <-> x = z) <-> -. E.zA.x(ph <-> x = z))
4		ax-1 6	. . . . . . . . . . 11 |- (-. A.x(ph <-> x = z) -> (z = z -> -. A.x(ph <-> x = z)))
5		eqidd 2354	. . . . . . . . . . 11 |- (-. A.x(ph <-> x = z) -> z = z)
6	4, 5	impbid1 194	. . . . . . . . . 10 |- (-. A.x(ph <-> x = z) -> (z = z <-> -. A.x(ph <-> x = z)))
7	6	con2bid 319	. . . . . . . . 9 |- (-. A.x(ph <-> x = z) -> (A.x(ph <-> x = z) <-> -. z = z))
8	7	alimi 1559	. . . . . . . 8 |- (A.z -. A.x(ph <-> x = z) -> A.z(A.x(ph <-> x = z) <-> -. z = z))
9		abbi 2464	. . . . . . . 8 |- (A.z(A.x(ph <-> x = z) <-> -. z = z) <-> {z | A.x(ph <-> x = z)} = {z | -. z = z})
10	8, 9	sylib 188	. . . . . . 7 |- (A.z -. A.x(ph <-> x = z) -> {z | A.x(ph <-> x = z)} = {z | -. z = z})
11		dfnul2 3553	. . . . . . 7 |- (/) = {z | -. z = z}
12	10, 11	syl6eqr 2403	. . . . . 6 |- (A.z -. A.x(ph <-> x = z) -> {z | A.x(ph <-> x = z)} = (/))
13	3, 12	sylbir 204	. . . . 5 |- (-. E.zA.x(ph <-> x = z) -> {z | A.x(ph <-> x = z)} = (/))
14	13	unieqd 3903	. . . 4 |- (-. E.zA.x(ph <-> x = z) -> U.{z | A.x(ph <-> x = z)} = U.(/))
15		uni0 3919	. . . 4 |- U.(/) = (/)
16	14, 15	syl6eq 2401	. . 3 |- (-. E.zA.x(ph <-> x = z) -> U.{z | A.x(ph <-> x = z)} = (/))
17	2, 16	syl5eq 2397	. 2 |- (-. E.zA.x(ph <-> x = z) -> (iotaxph) = (/))
18	1, 17	sylnbi 297	1 |- (-. E!xph -> (iotaxph) = (/))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642  E!weu 2204  {cab 2339  (/)c0 3551  U.cuni 3892  iotacio 4338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  iotassuni  4356  iotaex  4357  dfiota3  4371  dfiota4  4373  fvprc  5326