Theorem euuni 3807

Description: If ph is true for exactly one x, then U.{x | ph} is a way to express "the unique element such that ph is true." Some books use a special symbol such as iota to denote "the unique element such that."

Assertion

Ref	Expression
euuni	|- (E!xph -> (ph <-> U.{x | ph} = x))

Proof of Theorem euuni

Step	Hyp	Ref	Expression
1		euabex 3514	. . . 4 |- (E!xph -> {x | ph} e. _V)
2		uniexg 3795	. . . 4 |- ({x | ph} e. _V -> U.{x | ph} e. _V)
3	1, 2	syl 12	. . 3 |- (E!xph -> U.{x | ph} e. _V)
4		eueq 2427	. . . 4 |- (U.{x | ph} e. _V <-> E!y y = U.{x | ph})
5		eqcom 1886	. . . . 5 |- (y = U.{x | ph} <-> U.{x | ph} = y)
6	5	eubii 1780	. . . 4 |- (E!y y = U.{x | ph} <-> E!yU.{x | ph} = y)
7		hbab1 1874	. . . . . . 7 |- (z e. {x | ph} -> A.x z e. {x | ph})
8	7	hbuni 3183	. . . . . 6 |- (z e. U.{x | ph} -> A.x z e. U.{x | ph})
9		ax-17 1317	. . . . . 6 |- (z e. y -> A.x z e. y)
10	8, 9	hbeq 1995	. . . . 5 |- (U.{x | ph} = y -> A.xU.{x | ph} = y)
11		ax-17 1317	. . . . 5 |- (U.{x | ph} = x -> A.yU.{x | ph} = x)
12		eqeq2 1893	. . . . 5 |- (y = x -> (U.{x | ph} = y <-> U.{x | ph} = x))
13	10, 11, 12	cbveu 1785	. . . 4 |- (E!yU.{x | ph} = y <-> E!xU.{x | ph} = x)
14	4, 6, 13	3bitri 194	. . 3 |- (U.{x | ph} e. _V <-> E!xU.{x | ph} = x)
15	3, 14	sylib 215	. 2 |- (E!xph -> E!xU.{x | ph} = x)
16		euabsn 3095	. . 3 |- (E!xph <-> E.x{x | ph} = {x})
17		visset 2295	. . . . . . . 8 |- x e. _V
18	17	snid 3069	. . . . . . 7 |- x e. {x}
19		eleq2 1958	. . . . . . 7 |- ({x | ph} = {x} -> (x e. {x | ph} <-> x e. {x}))
20	18, 19	mpbiri 211	. . . . . 6 |- ({x | ph} = {x} -> x e. {x | ph})
21		abid 1873	. . . . . 6 |- (x e. {x | ph} <-> ph)
22	20, 21	sylib 215	. . . . 5 |- ({x | ph} = {x} -> ph)
23		unieq 3185	. . . . . 6 |- ({x | ph} = {x} -> U.{x | ph} = U.{x})
24	17	unisn 3193	. . . . . 6 |- U.{x} = x
25	23, 24	syl6eq 1944	. . . . 5 |- ({x | ph} = {x} -> U.{x | ph} = x)
26	22, 25	jca 310	. . . 4 |- ({x | ph} = {x} -> (ph /\ U.{x | ph} = x))
27	26	eximi 1387	. . 3 |- (E.x{x | ph} = {x} -> E.x(ph /\ U.{x | ph} = x))
28	16, 27	sylbi 216	. 2 |- (E!xph -> E.x(ph /\ U.{x | ph} = x))
29		eupickb 1836	. 2 |- ((E!xph /\ E!xU.{x | ph} = x /\ E.x(ph /\ U.{x | ph} = x)) -> (ph <-> U.{x | ph} = x))
30	15, 28, 29	mpd3an23 1193	1 |- (E!xph -> (ph <-> U.{x | ph} = x))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  _Vcvv 2292  {csn 3044  U.cuni 3177

This theorem is referenced by:  reuuni1 3808  reuuni1OLD 3809  iota1 5104

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-uni 3178

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