Theorem unisng 3761

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 3543	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	unieqd 3755	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴})
3		id 19	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2155	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴))
5		vex 2692	. . 3 ⊢ 𝑥 ∈ V
6	5	unisn 3760	. 2 ⊢ ∪ {𝑥} = 𝑥
7	4, 6	vtoclg 2749	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  {csn 3532  ∪ cuni 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-uni 3745

This theorem is referenced by:  dfnfc2  3762  unisucg  4344  unisn3  4374  opswapg  5033  funfvdm  5492  en2other2  7069