Theorem unisn 3718

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 3507	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	unieqi 3712	. 2 ⊢ ∪ {𝐴} = ∪ {𝐴, 𝐴}
3		unisn.1	. . 3 ⊢ 𝐴 ∈ V
4	3, 3	unipr 3716	. 2 ⊢ ∪ {𝐴, 𝐴} = (𝐴 ∪ 𝐴)
5		unidm 3185	. 2 ⊢ (𝐴 ∪ 𝐴) = 𝐴
6	2, 4, 5	3eqtri 2139	1 ⊢ ∪ {𝐴} = 𝐴

Syntax hints:   = wceq 1314   ∈ wcel 1463  Vcvv 2657   ∪ cun 3035  {csn 3493  {cpr 3494  ∪ cuni 3702

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659  df-un 3041  df-sn 3499  df-pr 3500  df-uni 3703

This theorem is referenced by:  unisng  3719  uniintsnr  3773  unisuc  4295  op1sta  4978  op2nda  4981  elxp4  4984  uniabio  5056  iotass  5063  en1bg  6648