Theorem bj-snglinv 35064

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 35061	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ∈ sngl 𝐴)
2	1	abbi2i 2879	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Syntax hints:   = wceq 1543   ∈ wcel 2112  {cab 2716  {csn 4558  sngl bj-csngl 35057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pr 5346

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-rex 3070  df-v 3425  df-dif 3887  df-un 3889  df-nul 4255  df-sn 4559  df-pr 4561  df-bj-sngl 35058

This theorem is referenced by:  bj-snglex  35065  bj-taginv  35078