Theorem bj-snglinv 37649

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 37646	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ∈ sngl 𝐴)
2	1	eqabi 2901	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {cab 2744  {csn 4594  sngl bj-csngl 37642

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rex 3093  df-v 3460  df-un 3913  df-sn 4595  df-pr 4597  df-bj-sngl 37643

This theorem is used by:  bj-snglex  37650  bj-taginv  37663