Theorem bj-snglinv 34284

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 34281	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ∈ sngl 𝐴)
2	1	abbi2i 2953	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Syntax hints:   = wceq 1533   ∈ wcel 2110  {cab 2799  {csn 4566  sngl bj-csngl 34277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4567  df-pr 4569  df-bj-sngl 34278

This theorem is referenced by:  bj-snglex  34285  bj-taginv  34298