Theorem rabsn 3791

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 2413	. . . . 5 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
2	1	pm5.32ri 619	. . . 4 ⊢ ((x ∈ A ∧ x = B) ↔ (B ∈ A ∧ x = B))
3	2	baib 871	. . 3 ⊢ (B ∈ A → ((x ∈ A ∧ x = B) ↔ x = B))
4	3	abbidv 2468	. 2 ⊢ (B ∈ A → {x ∣ (x ∈ A ∧ x = B)} = {x ∣ x = B})
5		df-rab 2624	. 2 ⊢ {x ∈ A ∣ x = B} = {x ∣ (x ∈ A ∧ x = B)}
6		df-sn 3742	. 2 ⊢ {B} = {x ∣ x = B}
7	4, 5, 6	3eqtr4g 2410	1 ⊢ (B ∈ A → {x ∈ A ∣ x = B} = {B})

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-rab 2624  df-sn 3742

This theorem is referenced by: (None)