Theorem unisng 2518

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.

Assertion

Ref	Expression
unisng	|- (A e. B -> U.{A} = A)

Proof of Theorem unisng

Step	Hyp	Ref	Expression
1		sneq 2417	. . . 4 |- (x = A -> {x} = {A})
2	1	unieqd 2512	. . 3 |- (x = A -> U.{x} = U.{A})
3		id 59	. . 3 |- (x = A -> x = A)
4	2, 3	eqeq12d 1489	. 2 |- (x = A -> (U.{x} = x <-> U.{A} = A))
5		visset 1813	. . 3 |- x e. V
6	5	unisn 2517	. 2 |- U.{x} = x
7	4, 6	vtoclg 1847	1 |- (A e. B -> U.{A} = A)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  {csn 2409  U.cuni 2503

This theorem is referenced by:  unisn2 2875  unisn3 2876  chsupsn 9312  oefil2 10567

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-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-uni 2504

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