Theorem unisn3 4859

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 4654	. . 3 ⊢ (𝐴 ∈ 𝐵 → {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = {𝐴})
2	1	unieqd 4850	. 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = ∪ {𝐴})
3		unisng 4857	. 2 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝐴} = 𝐴)
4	2, 3	eqtrd 2778	1 ⊢ (𝐴 ∈ 𝐵 → ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 = 𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {crab 3067  {csn 4558  ∪ cuni 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-pr 4561  df-uni 4837

This theorem is referenced by: (None)