Theorem reucl 2885

Description: Closure law for "the unique element in A such that ph."

Assertion

Ref	Expression
reucl	|- (E!x e. A ph -> U.{x e. A | ph} e. A)

Distinct variable group:   x,A

Proof of Theorem reucl

Step	Hyp	Ref	Expression
1		eusn 2446	. . 3 |- (E!x(x e. A /\ ph) <-> E.x{x | (x e. A /\ ph)} = {x})
2		hbab1 1466	. . . . . 6 |- (y e. {x | (x e. A /\ ph)} -> A.x y e. {x | (x e. A /\ ph)})
3	2	hbuni 2509	. . . . 5 |- (y e. U.{x | (x e. A /\ ph)} -> A.x y e. U.{x | (x e. A /\ ph)})
4		ax-17 971	. . . . 5 |- (y e. A -> A.x y e. A)
5	3, 4	hbel 1566	. . . 4 |- (U.{x | (x e. A /\ ph)} e. A -> A.xU.{x | (x e. A /\ ph)} e. A)
6		unieq 2510	. . . . . 6 |- ({x | (x e. A /\ ph)} = {x} -> U.{x | (x e. A /\ ph)} = U.{x})
7		visset 1813	. . . . . . 7 |- x e. V
8	7	unisn 2517	. . . . . 6 |- U.{x} = x
9	6, 8	syl6req 1524	. . . . 5 |- ({x | (x e. A /\ ph)} = {x} -> x = U.{x | (x e. A /\ ph)})
10	7	snid 2435	. . . . . . . 8 |- x e. {x}
11		eleq2 1535	. . . . . . . 8 |- ({x | (x e. A /\ ph)} = {x} -> (x e. {x | (x e. A /\ ph)} <-> x e. {x}))
12	10, 11	mpbiri 194	. . . . . . 7 |- ({x | (x e. A /\ ph)} = {x} -> x e. {x | (x e. A /\ ph)})
13		abid 1465	. . . . . . 7 |- (x e. {x | (x e. A /\ ph)} <-> (x e. A /\ ph))
14	12, 13	sylib 198	. . . . . 6 |- ({x | (x e. A /\ ph)} = {x} -> (x e. A /\ ph))
15	14	pm3.26d 321	. . . . 5 |- ({x | (x e. A /\ ph)} = {x} -> x e. A)
16	9, 15	eqeltrrd 1549	. . . 4 |- ({x | (x e. A /\ ph)} = {x} -> U.{x | (x e. A /\ ph)} e. A)
17	5, 16	19.23ai 1064	. . 3 |- (E.x{x | (x e. A /\ ph)} = {x} -> U.{x | (x e. A /\ ph)} e. A)
18	1, 17	sylbi 199	. 2 |- (E!x(x e. A /\ ph) -> U.{x | (x e. A /\ ph)} e. A)
19		df-reu 1651	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
20		df-rab 1652	. . . 4 |- {x e. A | ph} = {x | (x e. A /\ ph)}
21	20	unieqi 2511	. . 3 |- U.{x e. A | ph} = U.{x | (x e. A /\ ph)}
22	21	eleq1i 1537	. 2 |- (U.{x e. A | ph} e. A <-> U.{x | (x e. A /\ ph)} e. A)
23	18, 19, 22	3imtr4 219	1 |- (E!x e. A ph -> U.{x e. A | ph} e. A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380  {cab 1463  E!wreu 1647  {crab 1648  {csn 2409  U.cuni 2503

This theorem is referenced by:  reuuni3 2886  reuuni4 2887  reucl2 2888  reuuniss 2889  reuuniss2 2891  reuunixfr 2906  wereucl 2946  supcl 4579  aceq6a 4741  subcl 5366  divcl 5710  uzwo3lem2 6217  flclt 6226  reclt 6757  imclt 6758  isumclt 7209  grpidcl 8059  grpinvcl 8068  minveccl 8584  spwcl 8660  axpjclt 9240  cnlnadjlem3 10002  cnlnadjlem4 10003  adjbdlnt 10016

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-clab 1464  df-cleq 1469  df-clel 1472  df-reu 1651  df-rab 1652  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-uni 2504

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