Theorem rp-abid 43417

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 3620	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎)
2	1	eqabi 2866	1 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}

Syntax hints:   = wceq 1541  {cab 2709  ∃wrex 3056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rex 3057

This theorem is referenced by:  oaun2  43420  oaun3  43421