Theorem euuni 2844

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 2735	. . . 4 |- (E!xph -> {x | ph} e. V)
2		uniexg 2835	. . . 4 |- ({x | ph} e. V -> U.{x | ph} e. V)
3	1, 2	syl 10	. . 3 |- (E!xph -> U.{x | ph} e. V)
4		eueq 1888	. . . 4 |- (U.{x | ph} e. V <-> E!y y = U.{x | ph})
5		eqcom 1453	. . . . 5 |- (y = U.{x | ph} <-> U.{x | ph} = y)
6	5	eubii 1364	. . . 4 |- (E!y y = U.{x | ph} <-> E!yU.{x | ph} = y)
7		hbab1 1443	. . . . . . 7 |- (z e. {x | ph} -> A.x z e. {x | ph})
8	7	hbuni 2477	. . . . . 6 |- (z e. U.{x | ph} -> A.x z e. U.{x | ph})
9		ax-17 1190	. . . . . 6 |- (z e. y -> A.x z e. y)
10	8, 9	hbeq 1541	. . . . 5 |- (U.{x | ph} = y -> A.xU.{x | ph} = y)
11		ax-17 1190	. . . . 5 |- (U.{x | ph} = x -> A.yU.{x | ph} = x)
12		eqeq2 1460	. . . . 5 |- (y = x -> (U.{x | ph} = y <-> U.{x | ph} = x))
13	10, 11, 12	cbveu 1368	. . . 4 |- (E!yU.{x | ph} = y <-> E!xU.{x | ph} = x)
14	4, 6, 13	3bitr 177	. . 3 |- (U.{x | ph} e. V <-> E!xU.{x | ph} = x)
15	3, 14	sylib 198	. 2 |- (E!xph -> E!xU.{x | ph} = x)
16		eusn 2416	. . 3 |- (E!xph <-> E.x{x | ph} = {x})
17		visset 1788	. . . . . . . 8 |- x e. V
18	17	snid 2406	. . . . . . 7 |- x e. {x}
19		eleq2 1511	. . . . . . 7 |- ({x | ph} = {x} -> (x e. {x | ph} <-> x e. {x}))
20	18, 19	mpbiri 194	. . . . . 6 |- ({x | ph} = {x} -> x e. {x | ph})
21		abid 1442	. . . . . 6 |- (x e. {x | ph} <-> ph)
22	20, 21	sylib 198	. . . . 5 |- ({x | ph} = {x} -> ph)
23		unieq 2478	. . . . . 6 |- ({x | ph} = {x} -> U.{x | ph} = U.{x})
24	17	unisn 2485	. . . . . 6 |- U.{x} = x
25	23, 24	syl6eq 1499	. . . . 5 |- ({x | ph} = {x} -> U.{x | ph} = x)
26	22, 25	jca 288	. . . 4 |- ({x | ph} = {x} -> (ph /\ U.{x | ph} = x))
27	26	19.22i 1016	. . 3 |- (E.x{x | ph} = {x} -> E.x(ph /\ U.{x | ph} = x))
28	16, 27	sylbi 199	. 2 |- (E!xph -> E.x(ph /\ U.{x | ph} = x))
29		eupickb 1412	. 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 914	1 |- (E!xph -> (ph <-> U.{x | ph} = x))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  E.wex 956   = wceq 1099   e. wcel 1105  E!weu 1357  {cab 1440  Vcvv 1786  {csn 2380  U.cuni 2471

This theorem is referenced by:  reuuni1 2845

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-uni 2472

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