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Theorem bj-snglex 33800
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 3428 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 pweq 4425 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
32eximi 1797 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4 bj-snglss 33797 . . . . . 6 sngl 𝐴 ⊆ 𝒫 𝐴
5 sseq2 3884 . . . . . 6 (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
64, 5mpbiri 250 . . . . 5 (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
76eximi 1797 . . . 4 (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8 vpwex 5131 . . . . . 6 𝒫 𝑥 ∈ V
98ssex 5081 . . . . 5 (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
109exlimiv 1889 . . . 4 (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
113, 7, 103syl 18 . . 3 (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V)
121, 11sylbi 209 . 2 (𝐴 ∈ V → sngl 𝐴 ∈ V)
13 bj-snglinv 33799 . . 3 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14 bj-snsetex 33790 . . 3 (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
1513, 14syl5eqel 2871 . 2 (sngl 𝐴 ∈ V → 𝐴 ∈ V)
1612, 15impbii 201 1 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1507  wex 1742  wcel 2050  {cab 2759  Vcvv 3416  wss 3830  𝒫 cpw 4422  {csn 4441  sngl bj-csngl 33792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-rex 3095  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-pw 4424  df-sn 4442  df-pr 4444  df-bj-sngl 33793
This theorem is referenced by:  bj-tagex  33814
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