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Theorem bj-snglex 36452
Description: A class is a set if and only if its singletonization is a set. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snglex (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)

Proof of Theorem bj-snglex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isset 3484 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 pweq 4617 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
32eximi 1830 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4 bj-snglss 36449 . . . . . 6 sngl 𝐴 ⊆ 𝒫 𝐴
5 sseq2 4006 . . . . . 6 (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
64, 5mpbiri 258 . . . . 5 (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
76eximi 1830 . . . 4 (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8 vpwex 5377 . . . . . 6 𝒫 𝑥 ∈ V
98ssex 5321 . . . . 5 (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
109exlimiv 1926 . . . 4 (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
113, 7, 103syl 18 . . 3 (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V)
121, 11sylbi 216 . 2 (𝐴 ∈ V → sngl 𝐴 ∈ V)
13 bj-snglinv 36451 . . 3 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14 bj-snsetex 36442 . . 3 (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
1513, 14eqeltrid 2833 . 2 (sngl 𝐴 ∈ V → 𝐴 ∈ V)
1612, 15impbii 208 1 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wex 1774  wcel 2099  {cab 2705  Vcvv 3471  wss 3947  𝒫 cpw 4603  {csn 4629  sngl bj-csngl 36444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632  df-bj-sngl 36445
This theorem is referenced by:  bj-tagex  36466
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