Theorem unisng 3909

Description: A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53. (Contributed by NM, 13-Aug-2002.)

Assertion

Ref	Expression
unisng	⊢ (A ∈ V → ∪{A} = A)

Proof of Theorem unisng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3745	. . . 4 ⊢ (x = A → {x} = {A})
2	1	unieqd 3903	. . 3 ⊢ (x = A → ∪{x} = ∪{A})
3		id 19	. . 3 ⊢ (x = A → x = A)
4	2, 3	eqeq12d 2367	. 2 ⊢ (x = A → (∪{x} = x ↔ ∪{A} = A))
5		vex 2863	. . 3 ⊢ x ∈ V
6	5	unisn 3908	. 2 ⊢ ∪{x} = x
7	4, 6	vtoclg 2915	1 ⊢ (A ∈ V → ∪{A} = A)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {csn 3738  ∪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:  dfnfc2  3910  dfiota4  4373