Theorem bj-snglinv 32960

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 32957	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ∈ sngl 𝐴)
2	1	abbi2i 2738	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}

Syntax hints:   = wceq 1483   ∈ wcel 1990  {cab 2608  {csn 4177  sngl bj-csngl 32953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-bj-sngl 32954

This theorem is referenced by:  bj-snglex  32961  bj-taginv  32974