Theorem rabsn 3650

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 2233	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	pm5.32ri 452	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵))
3	2	baib 914	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
4	3	abbidv 2288	. 2 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)} = {𝑥 ∣ 𝑥 = 𝐵})
5		df-rab 2457	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)}
6		df-sn 3589	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
7	4, 5, 6	3eqtr4g 2228	1 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵})

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141  {cab 2156  {crab 2452  {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rab 2457  df-sn 3589

This theorem is referenced by:  unisn3  4430