Theorem unisng 3194

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 3054	. . . 4 |- (x = A -> {x} = {A})
2	1	unieqd 3188	. . 3 |- (x = A -> U.{x} = U.{A})
3		id 73	. . 3 |- (x = A -> x = A)
4	2, 3	eqeq12d 1899	. 2 |- (x = A -> (U.{x} = x <-> U.{A} = A))
5		visset 2295	. . 3 |- x e. _V
6	5	unisn 3193	. 2 |- U.{x} = x
7	4, 6	vtoclg 2346	1 |- (A e. B -> U.{A} = A)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {csn 3044  U.cuni 3177

This theorem is referenced by:  unisn2 3799  unisn3 3800  fvopab6 4757  indistop 8918  oefil2 10275  extbas2 10292  chsupsn 10945  cptclsscpt 15432  fnejoin2 15531  ufileu 15573  filufint 15574  uffixfr 15575  flimcls 15588

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-uni 3178

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