Theorem rabsn 3772

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 2301	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	pm5.32ri 459	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵))
3	2	baib 931	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
4	3	abbidv 2358	. 2 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)} = {𝑥 ∣ 𝑥 = 𝐵})
5		df-rab 2537	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵)}
6		df-sn 3711	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
7	4, 5, 6	3eqtr4g 2296	1 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  {crab 2532  {csn 3705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rab 2537  df-sn 3711

This theorem is referenced by:  unisn3  4586  ssenneg  11258