Theorem unisng 3822

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 3600	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	unieqd 3816	. . 3 ⊢ (𝑥 = 𝐴 → ∪ {𝑥} = ∪ {𝐴})
3		id 19	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2190	. 2 ⊢ (𝑥 = 𝐴 → (∪ {𝑥} = 𝑥 ↔ ∪ {𝐴} = 𝐴))
5		vex 2738	. . 3 ⊢ 𝑥 ∈ V
6	5	unisn 3821	. 2 ⊢ ∪ {𝑥} = 𝑥
7	4, 6	vtoclg 2795	1 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2146  {csn 3589  ∪ cuni 3805

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-uni 3806

This theorem is referenced by:  dfnfc2  3823  unisucg  4408  unisn3  4439  opswapg  5107  funfvdm  5571  en2other2  7185