Theorem rp-abid 43830

Description: Two ways to express a class. (Contributed by RP, 13-Feb-2025.)

Assertion

Ref	Expression
rp-abid	⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}

Distinct variable group:   𝐴,𝑎,𝑥

Proof of Theorem rp-abid

Step	Hyp	Ref	Expression
1		clel5 3610	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎)
2	1	eqabi 2875	1 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}

Syntax hints:   = wceq 1547  {cab 2718  ∃wrex 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rex 3065

This theorem is referenced by:  oaun2  43833  oaun3  43834