Theorem unisn 3908

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 3748	. . 3 ⊢ {A} = {A, A}
2	1	unieqi 3902	. 2 ⊢ ∪{A} = ∪{A, A}
3		unisn.1	. . 3 ⊢ A ∈ V
4	3, 3	unipr 3906	. 2 ⊢ ∪{A, A} = (A ∪ A)
5		unidm 3408	. 2 ⊢ (A ∪ A) = A
6	2, 4, 5	3eqtri 2377	1 ⊢ ∪{A} = A

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208  {csn 3738  {cpr 3739  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893

This theorem is referenced by:  unisng  3909  uniintsn  3964  pw1eqadj  4333  uniabio  4350  nnadjoin  4521  op1sta  5073  opswap  5075  op2nda  5077  funfv  5376  pw1fnval  5852  pw1fnf1o  5856  brtcfn  6247