Theorem rabsn 3459

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 2141	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	pm5.32ri 442	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵))
3	2	baib 861	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
4	3	abbidv 2196	. 2 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)} = {𝑥 ∣ 𝑥 = 𝐵})
5		df-rab 2357	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)}
6		df-sn 3404	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
7	4, 5, 6	3eqtr4g 2138	1 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵})

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  {cab 2067  {crab 2352  {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-rab 2357  df-sn 3404

This theorem is referenced by:  unisn3  4198