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Theorem bj-sngltag 33302
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 33301 . 2 ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
2 df-bj-tag 33294 . . . 4 tag 𝐵 = (sngl 𝐵 ∪ {∅})
32eleq2i 2842 . . 3 ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅}))
4 elun 3904 . . . 4 ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}))
5 idd 24 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵))
6 elsni 4333 . . . . . 6 ({𝐴} ∈ {∅} → {𝐴} = ∅)
7 snprc 4389 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
8 elex 3364 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
98pm2.24d 148 . . . . . . 7 (𝐴𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵))
107, 9syl5bir 233 . . . . . 6 (𝐴𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵))
116, 10syl5 34 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵))
125, 11jaod 848 . . . 4 (𝐴𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵))
134, 12syl5bi 232 . . 3 (𝐴𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵))
143, 13syl5bi 232 . 2 (𝐴𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵))
151, 14impbid2 216 1 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 836   = wceq 1631  wcel 2145  Vcvv 3351  cun 3721  c0 4063  {csn 4316  sngl bj-csngl 33284  tag bj-ctag 33293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4317  df-bj-tag 33294
This theorem is referenced by:  bj-tagcg  33304  bj-taginv  33305
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