Theorem bj-taginv 36999

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 36985	. 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2		bj-sngltag 36996	. . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
3	2	elv 3439	. . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
4	3	abbii 2797	. 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
5	1, 4	eqtri 2753	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2110  {cab 2708  Vcvv 3434  {csn 4574  sngl bj-csngl 36978  tag bj-ctag 36987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-v 3436  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4282  df-sn 4575  df-pr 4577  df-bj-sngl 36979  df-bj-tag 36988

This theorem is referenced by:  bj-projval  37009