Theorem unisn 2521

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.

Hypothesis

Ref	Expression
unisn.1	⊢ A ∈ V

Assertion

Ref	Expression
unisn	⊢ ∪{A} = A

Proof of Theorem unisn

Step	Hyp	Ref	Expression
1		dfsn2 2424	. . 3 ⊢ {A} = {A, A}
2	1	unieqi 2515	. 2 ⊢ ∪{A} = ∪{A, A}
3		unisn.1	. . 3 ⊢ A ∈ V
4	3, 3	unipr 2519	. 2 ⊢ ∪{A, A} = (A ∪ A)
5		unidm 2178	. 2 ⊢ (A ∪ A) = A
6	2, 4, 5	3eqtr 1502	1 ⊢ ∪{A} = A

Syntax hints:   = wceq 958   ∈ wcel 960  Vcvv 1814   ∪ cun 2048  {csn 2413  {cpr 2414  ∪cuni 2507

This theorem is referenced by:  unisng 2522  unidif0 2744  euuni 2887  reucl 2891  rabsnt 2900  reuunisn 2901  unisuc 3052  onuninsuc 3114  op1sta 3454  unixp0 3524  fvex 3738  funfv 3776  ecqs 4303  xpcomen 4445  unifiOLD 4570  subtop 7643  sn0top 7644  indistop 7645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-uni 2508

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