Theorem unisn 3914

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		dfsn2 3687	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 3908	. 2 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3		unisn.1	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	unipr 3912	. 2 ⊢ ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴)
5		unidm 3352	. 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴
6	2, 4, 5	3eqtri 2256	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199  {csn 3673  {cpr 3674  ∪ cuni 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-uni 3899

This theorem is referenced by:  unisng  3915  uniintsnr  3969  unisuc  4516  op1sta  5225  op2nda  5228  elxp4  5231  uniabio  5304  iotass  5311  en1bg  7017  zrhval2  14695