Theorem bj-taginv 32974

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 32960	. 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2		vex 3203	. . . 4 ⊢ 𝑥 ∈ V
3		bj-sngltag 32971	. . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
4	2, 3	ax-mp 5	. . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
5	4	abbii 2739	. 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
6	1, 5	eqtri 2644	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

Syntax hints:   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200  {csn 4177  sngl bj-csngl 32953  tag bj-ctag 32962

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-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-bj-sngl 32954  df-bj-tag 32963

This theorem is referenced by:  bj-projval  32984