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Theorem bj-sngltag 33463
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 33462 . 2 ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ tag 𝐵)
2 df-bj-tag 33455 . . . 4 tag 𝐵 = (sngl 𝐵 ∪ {∅})
32eleq2i 2870 . . 3 ({𝐴} ∈ tag 𝐵 ↔ {𝐴} ∈ (sngl 𝐵 ∪ {∅}))
4 elun 3951 . . . 4 ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) ↔ ({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}))
5 idd 24 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 → {𝐴} ∈ sngl 𝐵))
6 elsni 4385 . . . . . 6 ({𝐴} ∈ {∅} → {𝐴} = ∅)
7 snprc 4442 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
8 elex 3400 . . . . . . . 8 (𝐴𝑉𝐴 ∈ V)
98pm2.24d 149 . . . . . . 7 (𝐴𝑉 → (¬ 𝐴 ∈ V → {𝐴} ∈ sngl 𝐵))
107, 9syl5bir 235 . . . . . 6 (𝐴𝑉 → ({𝐴} = ∅ → {𝐴} ∈ sngl 𝐵))
116, 10syl5 34 . . . . 5 (𝐴𝑉 → ({𝐴} ∈ {∅} → {𝐴} ∈ sngl 𝐵))
125, 11jaod 886 . . . 4 (𝐴𝑉 → (({𝐴} ∈ sngl 𝐵 ∨ {𝐴} ∈ {∅}) → {𝐴} ∈ sngl 𝐵))
134, 12syl5bi 234 . . 3 (𝐴𝑉 → ({𝐴} ∈ (sngl 𝐵 ∪ {∅}) → {𝐴} ∈ sngl 𝐵))
143, 13syl5bi 234 . 2 (𝐴𝑉 → ({𝐴} ∈ tag 𝐵 → {𝐴} ∈ sngl 𝐵))
151, 14impbid2 218 1 (𝐴𝑉 → ({𝐴} ∈ sngl 𝐵 ↔ {𝐴} ∈ tag 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wo 874   = wceq 1653  wcel 2157  Vcvv 3385  cun 3767  c0 4115  {csn 4368  sngl bj-csngl 33445  tag bj-ctag 33454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-sn 4369  df-bj-tag 33455
This theorem is referenced by:  bj-tagcg  33465  bj-taginv  33466
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