Theorem rabsn 4654

Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.) (Proof shortened by AV, 26-Aug-2022.)

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 2826	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
2	1	pm5.32ri 575	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝐵 ∈ 𝐴 ∧ 𝑥 = 𝐵))
3	2	baib 535	. . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
4	3	alrimiv 1931	. 2 ⊢ (𝐵 ∈ 𝐴 → ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
5		rabeqsn 4599	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵} ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝐵) ↔ 𝑥 = 𝐵))
6	4, 5	sylibr 233	1 ⊢ (𝐵 ∈ 𝐴 → {𝑥 ∈ 𝐴 ∣ 𝑥 = 𝐵} = {𝐵})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  {crab 3067  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-sn 4559

This theorem is referenced by:  unisn3  4859  sylow3lem6  19152  lineunray  34376  pmapat  37704  dia0  38993  nzss  41824  lco0  45656