Theorem unisn 3821

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 3603	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 3815	. 2 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3		unisn.1	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	unipr 3819	. 2 ⊢ ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴)
5		unidm 3276	. 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴
6	2, 4, 5	3eqtri 2200	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   = wceq 1353   ∈ wcel 2146  Vcvv 2735   ∪ cun 3125  {csn 3589  {cpr 3590  ∪ cuni 3805

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-uni 3806

This theorem is referenced by:  unisng  3822  uniintsnr  3876  unisuc  4407  op1sta  5102  op2nda  5105  elxp4  5108  uniabio  5180  iotass  5187  en1bg  6790