Theorem rp-abid 44583

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 3622	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎)
2	1	eqabi 2899	1 ⊢ 𝐴 = {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = 𝑎}

This proof depends on syntax axioms:   = wceq 1570  {cab 2741  ∃wrex 3090

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091

This theorem is used by:  oaun2  44586  oaun3  44587