Theorem reuunisn 2895

Description: A restricted class abstraction with a unique member can be expressed as a singleton.

Assertion

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

Proof of Theorem reuunisn

Step	Hyp	Ref	Expression
1		reusn 2892	. 2 |- (E!x e. A ph <-> E.y{x e. A | ph} = {y})
2		unieq 2510	. . . . . 6 |- ({x e. A | ph} = {y} -> U.{x e. A | ph} = U.{y})
3		visset 1813	. . . . . . 7 |- y e. V
4	3	unisn 2517	. . . . . 6 |- U.{y} = y
5	2, 4	syl6eq 1523	. . . . 5 |- ({x e. A | ph} = {y} -> U.{x e. A | ph} = y)
6	5	sneqd 2419	. . . 4 |- ({x e. A | ph} = {y} -> {U.{x e. A | ph}} = {y})
7		eqtr3t 1494	. . . 4 |- (({x e. A | ph} = {y} /\ {U.{x e. A | ph}} = {y}) -> {x e. A | ph} = {U.{x e. A | ph}})
8	6, 7	mpdan 704	. . 3 |- ({x e. A | ph} = {y} -> {x e. A | ph} = {U.{x e. A | ph}})
9	8	19.23aiv 1295	. 2 |- (E.y{x e. A | ph} = {y} -> {x e. A | ph} = {U.{x e. A | ph}})
10	1, 9	sylbi 199	1 |- (E!x e. A ph -> {x e. A | ph} = {U.{x e. A | ph}})

Syntax hints:   -> wi 3   = wceq 956  E.wex 980  E!wreu 1647  {crab 1648  {csn 2409  U.cuni 2503

This theorem is referenced by:  pjspansnt 9500

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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