Theorem iunid 2598

Description: An indexed union of singletons recovers the index set.

Assertion

Ref	Expression
iunid	|- U_x e. A {x} = A

Distinct variable group:   x,A

Proof of Theorem iunid

Step	Hyp	Ref	Expression
1		eliun 2565	. . 3 |- (y e. U_x e. A {x} <-> E.x e. A y e. {x})
2		df-rex 1647	. . . 4 |- (E.x e. A y e. {x} <-> E.x(x e. A /\ y e. {x}))
3		ancom 435	. . . . . 6 |- ((x e. A /\ y e. {x}) <-> (y e. {x} /\ x e. A))
4		elsn 2417	. . . . . . . 8 |- (y e. {x} <-> y = x)
5		equcom 1127	. . . . . . . 8 |- (y = x <-> x = y)
6	4, 5	bitr 173	. . . . . . 7 |- (y e. {x} <-> x = y)
7	6	anbi1i 481	. . . . . 6 |- ((y e. {x} /\ x e. A) <-> (x = y /\ x e. A))
8	3, 7	bitr 173	. . . . 5 |- ((x e. A /\ y e. {x}) <-> (x = y /\ x e. A))
9	8	exbii 1049	. . . 4 |- (E.x(x e. A /\ y e. {x}) <-> E.x(x = y /\ x e. A))
10		ax-17 969	. . . . 5 |- (y e. A -> A.x y e. A)
11		eleq1 1531	. . . . 5 |- (x = y -> (x e. A <-> y e. A))
12	10, 11	equsex 1150	. . . 4 |- (E.x(x = y /\ x e. A) <-> y e. A)
13	2, 9, 12	3bitr 177	. . 3 |- (E.x e. A y e. {x} <-> y e. A)
14	1, 13	bitr 173	. 2 |- (y e. U_x e. A {x} <-> y e. A)
15	14	eqriv 1472	1 |- U_x e. A {x} = A

Syntax hints:   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  E.wrex 1643  {csn 2405  U_ciun 2561

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rex 1647  df-v 1808  df-sn 2408  df-iun 2563

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