Theorem unisn 3907

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 3747	. . 3 ⊢ {A} = {A, A}
2	1	unieqi 3901	. 2 ⊢ ∪{A} = ∪{A, A}
3		unisn.1	. . 3 ⊢ A ∈ V
4	3, 3	unipr 3905	. 2 ⊢ ∪{A, A} = (A ∪ A)
5		unidm 3407	. 2 ⊢ (A ∪ A) = A
6	2, 4, 5	3eqtri 2377	1 ⊢ ∪{A} = A

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207  {csn 3737  {cpr 3738  ∪cuni 3891

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-sn 3741  df-pr 3742  df-uni 3892

This theorem is referenced by:  unisng  3908  uniintsn  3963  pw1eqadj  4332  uniabio  4349  nnadjoin  4520  op1sta  5072  opswap  5074  op2nda  5076  funfv  5375  pw1fnval  5851  pw1fnf1o  5855  brtcfn  6246