Theorem bj-snglinv 37146

Description: Inverse of singletonization. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-snglinv	⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-snglinv

Step	Hyp	Ref	Expression
1		bj-snglc 37143	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ∈ sngl 𝐴)
2	1	eqabi 2870	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2713  {csn 4579  sngl bj-csngl 37139

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-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rex 3060  df-v 3441  df-un 3905  df-sn 4580  df-pr 4582  df-bj-sngl 37140

This theorem is referenced by:  bj-snglex  37147  bj-taginv  37160