Theorem unisn 2517

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

Hypothesis

Ref	Expression
unisn.1	|- A e. V

Assertion

Ref	Expression
unisn	|- U.{A} = A

Proof of Theorem unisn

Step	Hyp	Ref	Expression
1		dfsn2 2420	. . 3 |- {A} = {A, A}
2	1	unieqi 2511	. 2 |- U.{A} = U.{A, A}
3		unisn.1	. . 3 |- A e. V
4	3, 3	unipr 2515	. 2 |- U.{A, A} = (A u. A)
5		unidm 2175	. 2 |- (A u. A) = A
6	2, 4, 5	3eqtr 1499	1 |- U.{A} = A

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045  {csn 2409  {cpr 2410  U.cuni 2503

This theorem is referenced by:  unisng 2518  unidif0 2739  euuni 2881  reucl 2885  rabsnt 2894  reuunisn 2895  unisuc 3046  onuninsuc 3108  op1sta 3448  unixp0 3518  fvex 3732  funfv 3770  ecqs 4297  xpcomen 4439  unifiOLD 4557  subtop 7646  sn0top 7647  indistop 7648

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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