Theorem euuni 1954

Description: If ph is true for exactly one x, then U.{x | ph} is a way to express 'the unique element such that'. 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 1869	. . . 4 |- (E!xph -> {x | ph} e. V)
2		uniexg 1948	. . . 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 1427	. . . 4 |- (U.{x | ph} e. V <-> E!y y = U.{x | ph})
5		cleqcom 1103	. . . . 5 |- (y = U.{x | ph} <-> U.{x | ph} = y)
6	5	bieu 1014	. . . 4 |- (E!y y = U.{x | ph} <-> E!yU.{x | ph} = y)
7		hbab1 1095	. . . . . . 7 |- (z e. {x | ph} -> A.x z e. {x | ph})
8	7	hbuni 1925	. . . . . 6 |- (z e. U.{x | ph} -> A.x z e. U.{x | ph})
9		ax-17 925	. . . . . 6 |- (z e. y -> A.x z e. y)
10	8, 9	hbeq 1171	. . . . 5 |- (U.{x | ph} = y -> A.xU.{x | ph} = y)
11		ax-17 925	. . . . 5 |- (U.{x | ph} = x -> A.yU.{x | ph} = x)
12		cleq2 1110	. . . . 5 |- (y = x -> (U.{x | ph} = y <-> U.{x | ph} = x))
13	10, 11, 12	cbveu 1018	. . . 4 |- (E!yU.{x | ph} = y <-> E!xU.{x | ph} = x)
14	4, 6, 13	3bitr 155	. . 3 |- (U.{x | ph} e. V <-> E!xU.{x | ph} = x)
15	3, 14	sylib 173	. 2 |- (E!xph -> E!xU.{x | ph} = x)
16		eusn 1913	. . 3 |- (E!xph <-> E.x{x | ph} = {x})
17		visset 1350	. . . . . . . 8 |- x e. V
18	17	snid 1830	. . . . . . 7 |- x e. {x}
19		eleq2 1150	. . . . . . 7 |- ({x | ph} = {x} -> (x e. {x | ph} <-> x e. {x}))
20	18, 19	mpbiri 169	. . . . . 6 |- ({x | ph} = {x} -> x e. {x | ph})
21		abid 1094	. . . . . 6 |- (x e. {x | ph} <-> ph)
22	20, 21	sylib 173	. . . . 5 |- ({x | ph} = {x} -> ph)
23		unieq 1927	. . . . . 6 |- ({x | ph} = {x} -> U.{x | ph} = U.{x})
24	17	unisn 1932	. . . . . 6 |- U.{x} = x
25	23, 24	syl6eq 1140	. . . . 5 |- ({x | ph} = {x} -> U.{x | ph} = x)
26	22, 25	jca 236	. . . 4 |- ({x | ph} = {x} -> (ph /\ U.{x | ph} = x))
27	26	19.22i 723	. . 3 |- (E.x{x | ph} = {x} -> E.x(ph /\ U.{x | ph} = x))
28	16, 27	sylbi 174	. 2 |- (E!xph -> E.x(ph /\ U.{x | ph} = x))
29		eupickb 1056	. . . 4 |- ((E!xph /\ E!xU.{x | ph} = x /\ E.x(ph /\ U.{x | ph} = x)) -> (ph <-> U.{x | ph} = x))
30	29	3exp 611	. . 3 |- (E!xph -> (E!xU.{x | ph} = x -> (E.x(ph /\ U.{x | ph} = x) -> (ph <-> U.{x | ph} = x))))
31	30	imp3a 279	. 2 |- (E!xph -> ((E!xU.{x | ph} = x /\ E.x(ph /\ U.{x | ph} = x)) -> (ph <-> U.{x | ph} = x)))
32	15, 28, 31	mp2and 526	1 |- (E!xph -> (ph <-> U.{x | ph} = x))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   e. wel 803  E!weu 1007  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348  {csn 1808  U.cuni 1919

This theorem is referenced by:  reuuni1 1955

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-uni 1920

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