Theorem unisn 4926

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
unisn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
unisn	⊢ ∪ {𝐴} = 𝐴

Proof of Theorem unisn

Step	Hyp	Ref	Expression
1		unisn.1	. 2 ⊢ 𝐴 ∈ V
2		unisng 4925	. 2 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
3	1, 2	ax-mp 5	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626  ∪ cuni 4907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908

This theorem is referenced by:  unisnv  4927  unidif0  5360  op1sta  6245  op2nda  6248  opswap  6249  fvssunirnOLD  6940  funfv  6996  dffv2  7004  nlim1  8527  tc2  9782  cflim2  10303  fin1a2lem12  10451  acsmapd  18599  ghmqusnsglem1  19298  ghmquskerlem1  19301  pmtrprfval  19505  lspuni0  21008  lss0v  21015  zrhval2  21519  indistopon  23008  refun0  23523  qtopeu  23724  hmphindis  23805  filconn  23891  ufildr  23939  cnextfres1  24076  bday1s  27876  old1  27914  madeoldsuc  27923  zs12bday  28424  dimval  33651  dimvalfi  33652  locfinref  33840  pstmfval  33895  esumval  34047  esumpfinval  34076  esumpfinvalf  34077  prsiga  34132  carsggect  34320  indispconn  35239  onsucsuccmpi  36444  bj-nuliotaALT  37059  heiborlem3  37820  isomenndlem  46545  uniimaelsetpreimafv  47383