Theorem bj-taginv 37899

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 37885	. 2 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ sngl 𝐴}
2		bj-sngltag 37896	. . . 4 ⊢ (𝑥 ∈ V → ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴))
3	2	elv 3455	. . 3 ⊢ ({𝑥} ∈ sngl 𝐴 ↔ {𝑥} ∈ tag 𝐴)
4	3	abbii 2827	. 2 ⊢ {𝑥 ∣ {𝑥} ∈ sngl 𝐴} = {𝑥 ∣ {𝑥} ∈ tag 𝐴}
5	1, 4	eqtri 2783	1 ⊢ 𝐴 = {𝑥 ∣ {𝑥} ∈ tag 𝐴}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450  {csn 4583  sngl bj-csngl 37878  tag bj-ctag 37887

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-sn 4584  df-pr 4586  df-bj-sngl 37879  df-bj-tag 37888

This theorem is used by:  bj-projval  37909