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Theorem bj-sngltag 34192
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 34191 . 2 ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
2 df-bj-tag 34184 . . . 4 tag 𝐵 = (sngl 𝐵 ∪ {∅})
32eleq2i 2901 . . 3 ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅}))
4 elun 4122 . . . 4 ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}))
5 idd 24 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵))
6 elsni 4574 . . . . . 6 ({𝐴} ∈ {∅} → {𝐴} = ∅)
7 snprc 4645 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
8 elex 3510 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
98pm2.24d 154 . . . . . . 7 (𝐴𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵))
107, 9syl5bir 244 . . . . . 6 (𝐴𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵))
116, 10syl5 34 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵))
125, 11jaod 853 . . . 4 (𝐴𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵))
134, 12syl5bi 243 . . 3 (𝐴𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵))
143, 13syl5bi 243 . 2 (𝐴𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵))
151, 14impbid2 227 1 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wo 841   = wceq 1528  wcel 2105  Vcvv 3492  cun 3931  c0 4288  {csn 4557  sngl bj-csngl 34174  tag bj-ctag 34183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-sn 4558  df-bj-tag 34184
This theorem is referenced by:  bj-tagcg  34194  bj-taginv  34195
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