Theorem unisn3 4877

Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)

Assertion

Ref	Expression
unisn3	⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unisn3

Step	Hyp	Ref	Expression
1		rabsn 4671	. . 3 ⊢ (𝐴 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = {𝐴})
2	1	unieqd 4869	. 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = ∪ {𝐴})
3		unisng 4874	. 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝐴} = 𝐴)
4	2, 3	eqtrd 2766	1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  {crab 3395  {csn 4573  ∪ cuni 4856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-un 3902  df-ss 3914  df-sn 4574  df-pr 4576  df-uni 4857

This theorem is referenced by: (None)