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Theorem bj-sngltag 36519
Description: The singletonization and the tagging of a set contain the same singletons. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sngltag (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))

Proof of Theorem bj-sngltag
StepHypRef Expression
1 bj-sngltagi 36518 . 2 ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
2 df-bj-tag 36511 . . . 4 tag 𝐵 = (sngl 𝐵 ∪ {∅})
32eleq2i 2817 . . 3 ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅}))
4 elun 4141 . . . 4 ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}))
5 idd 24 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵))
6 elsni 4641 . . . . . 6 ({𝐴} ∈ {∅} → {𝐴} = ∅)
7 snprc 4717 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
8 elex 3482 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
98pm2.24d 151 . . . . . . 7 (𝐴𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵))
107, 9biimtrrid 242 . . . . . 6 (𝐴𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵))
116, 10syl5 34 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵))
125, 11jaod 857 . . . 4 (𝐴𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵))
134, 12biimtrid 241 . . 3 (𝐴𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵))
143, 13biimtrid 241 . 2 (𝐴𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵))
151, 14impbid2 225 1 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 845   = wceq 1533  wcel 2098  Vcvv 3463  cun 3937  c0 4318  {csn 4624  sngl bj-csngl 36501  tag bj-ctag 36510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-sn 4625  df-bj-tag 36511
This theorem is referenced by:  bj-tagcg  36521  bj-taginv  36522
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