Theorem bj-taginv 34293

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 34279	. 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2		bj-sngltag 34290	. . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
3	2	elv 3499	. . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
4	3	abbii 2886	. 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
5	1, 4	eqtri 2844	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Syntax hints:   ↔ wb 208   = wceq 1533   ∈ wcel 2110  {cab 2799  Vcvv 3494  {csn 4560  sngl bj-csngl 34272  tag bj-ctag 34281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4561  df-pr 4563  df-bj-sngl 34273  df-bj-tag 34282

This theorem is referenced by:  bj-projval  34303