Theorem euuni 2938

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 2823	. . . 4 |- (E!xph -> {x | ph} e. V)
2		uniexg 2927	. . . 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 1963	. . . 4 |- (U.{x | ph} e. V <-> E!y y = U.{x | ph})
5		eqcom 1524	. . . . 5 |- (y = U.{x | ph} <-> U.{x | ph} = y)
6	5	eubii 1429	. . . 4 |- (E!y y = U.{x | ph} <-> E!yU.{x | ph} = y)
7		hbab1 1512	. . . . . . 7 |- (z e. {x | ph} -> A.x z e. {x | ph})
8	7	hbuni 2563	. . . . . 6 |- (z e. U.{x | ph} -> A.x z e. U.{x | ph})
9		ax-17 1012	. . . . . 6 |- (z e. y -> A.x z e. y)
10	8, 9	hbeq 1612	. . . . 5 |- (U.{x | ph} = y -> A.xU.{x | ph} = y)
11		ax-17 1012	. . . . 5 |- (U.{x | ph} = x -> A.yU.{x | ph} = x)
12		eqeq2 1531	. . . . 5 |- (y = x -> (U.{x | ph} = y <-> U.{x | ph} = x))
13	10, 11, 12	cbveu 1433	. . . 4 |- (E!yU.{x | ph} = y <-> E!xU.{x | ph} = x)
14	4, 6, 13	3bitri 184	. . 3 |- (U.{x | ph} e. V <-> E!xU.{x | ph} = x)
15	3, 14	sylib 205	. 2 |- (E!xph -> E!xU.{x | ph} = x)
16		eusn 2498	. . 3 |- (E!xph <-> E.x{x | ph} = {x})
17		visset 1860	. . . . . . . 8 |- x e. V
18	17	snid 2487	. . . . . . 7 |- x e. {x}
19		eleq2 1582	. . . . . . 7 |- ({x | ph} = {x} -> (x e. {x | ph} <-> x e. {x}))
20	18, 19	mpbiri 201	. . . . . 6 |- ({x | ph} = {x} -> x e. {x | ph})
21		abid 1511	. . . . . 6 |- (x e. {x | ph} <-> ph)
22	20, 21	sylib 205	. . . . 5 |- ({x | ph} = {x} -> ph)
23		unieq 2564	. . . . . 6 |- ({x | ph} = {x} -> U.{x | ph} = U.{x})
24	17	unisn 2571	. . . . . 6 |- U.{x} = x
25	23, 24	syl6eq 1570	. . . . 5 |- ({x | ph} = {x} -> U.{x | ph} = x)
26	22, 25	jca 295	. . . 4 |- ({x | ph} = {x} -> (ph /\ U.{x | ph} = x))
27	26	19.22i 1081	. . 3 |- (E.x{x | ph} = {x} -> E.x(ph /\ U.{x | ph} = x))
28	16, 27	sylbi 206	. 2 |- (E!xph -> E.x(ph /\ U.{x | ph} = x))
29		eupickb 1478	. 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 930	1 |- (E!xph -> (ph <-> U.{x | ph} = x))

Syntax hints:   -> wi 3   <-> wb 153   /\ wa 230   = wceq 997   e. wcel 999  E.wex 1021  E!weu 1422  {cab 1509  Vcvv 1858  {csn 2461  U.cuni 2557

This theorem is referenced by:  reuuni1 2939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-uni 2558

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