Theorem unisng 3856

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 3633	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	unieqd 3850	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴})
3		id 19	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2211	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴))
5		vex 2766	. . 3 ⊢ 𝑥 ∈ V
6	5	unisn 3855	. 2 ⊢ ∪ {𝑥} = 𝑥
7	4, 6	vtoclg 2824	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  {csn 3622  ∪ cuni 3839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840

This theorem is referenced by:  dfnfc2  3857  unisucg  4449  unisn3  4480  opswapg  5156  funfvdm  5624  en2other2  7263  lspuni0  13980  lss0v  13986