Theorem rp-abid 43732

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 3621	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎)
2	1	eqabi 2872	1 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}

Syntax hints:   = wceq 1542  {cab 2715  ∃wrex 3062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063

This theorem is referenced by:  oaun2  43735  oaun3  43736