Theorem iotanul 5571

Description: Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one that satisfies . (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotanul

Proof of Theorem iotanul

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2286	. 2
2		dfiota2 5557	. . 3
3		alnex 1614	. . . . . 6
4		ax-1 6	. . . . . . . . . . 11
5		eqidd 2458	. . . . . . . . . . 11
6	4, 5	impbid1 203	. . . . . . . . . 10
7	6	con2bid 329	. . . . . . . . 9
8	7	alimi 1633	. . . . . . . 8
9		abbi 2588	. . . . . . . 8
10	8, 9	sylib 196	. . . . . . 7
11		dfnul2 3786	. . . . . . 7
12	10, 11	syl6eqr 2516	. . . . . 6
13	3, 12	sylbir 213	. . . . 5
14	13	unieqd 4259	. . . 4
15		uni0 4276	. . . 4
16	14, 15	syl6eq 2514	. . 3
17	2, 16	syl5eq 2510	. 2
18	1, 17	sylnbi 306	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  E.wex 1612  E!weu 2282  {cab 2442  c0 3784  U.cuni 4249  iotacio 5554

This theorem is referenced by:  iotassuni  5572  iotaex  5573  dfiota4  5584  csbiota  5585  tz6.12-2  5862  dffv3  5867  csbriota  6269  riotaund  6293  isf32lem9  8762  grpidval  15887  0g0  15890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-uni 4250  df-iota 5556