Theorem bj-taginv 37570

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

Assertion

Ref	Expression
bj-taginv	⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-taginv

Step	Hyp	Ref	Expression
1		bj-snglinv 37556	. 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2		bj-sngltag 37567	. . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
3	2	elv 3467	. . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
4	3	abbii 2837	. 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
5	1, 4	eqtri 2793	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Syntax hints:   ↔ wb 209   = wceq 1568   ∈ wcel 2150  {cab 2748  Vcvv 3462  {csn 4594  sngl bj-csngl 37549  tag bj-ctag 37558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rex 3097  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-sn 4595  df-pr 4597  df-bj-sngl 37550  df-bj-tag 37559

This theorem is referenced by:  bj-projval  37580