Theorem rp-abid 43340

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 3678	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎)
2	1	eqabi 2880	1 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}

Syntax hints:   = wceq 1537  {cab 2717  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077

This theorem is referenced by:  oaun2  43343  oaun3  43344