Theorem unisn 3903

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 3680	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 3897	. 2 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3		unisn.1	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	unipr 3901	. 2 ⊢ ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴)
5		unidm 3347	. 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴
6	2, 4, 5	3eqtri 2254	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195  {csn 3666  {cpr 3667  ∪ cuni 3887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3888

This theorem is referenced by:  unisng  3904  uniintsnr  3958  unisuc  4501  op1sta  5206  op2nda  5209  elxp4  5212  uniabio  5285  iotass  5292  en1bg  6942  zrhval2  14568