Theorem rabsn 4200

Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem rabsn

Step	Hyp	Ref	Expression
1		eleq1 2676	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	pm5.32ri 668	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵))
3	2	baib 942	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
4	3	abbidv 2728	. 2 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)} = {𝑥 ∣ 𝑥 = 𝐵})
5		df-rab 2905	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)}
6		df-sn 4126	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
7	4, 5, 6	3eqtr4g 2669	1 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-rab 2905  df-sn 4126

This theorem is referenced by:  unisn3  4389  sylow3lem6  17870  lineunray  31424  pmapat  34067  dia0  35359  nzss  37538  lco0  42010