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Theorem bj-snglex 36974
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 3494 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 pweq 4614 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
32eximi 1835 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4 bj-snglss 36971 . . . . . 6 sngl 𝐴 ⊆ 𝒫 𝐴
5 sseq2 4010 . . . . . 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 1835 . . . 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 1930 . . . 4 (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
113, 7, 103syl 18 . . 3 (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V)
121, 11sylbi 217 . 2 (𝐴 ∈ V → sngl 𝐴 ∈ V)
13 bj-snglinv 36973 . . 3 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14 bj-snsetex 36964 . . 3 (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
1513, 14eqeltrid 2845 . 2 (sngl 𝐴 ∈ V → 𝐴 ∈ V)
1612, 15impbii 209 1 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wex 1779  wcel 2108  {cab 2714  Vcvv 3480  wss 3951  𝒫 cpw 4600  {csn 4626  sngl bj-csngl 36966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629  df-bj-sngl 36967
This theorem is referenced by:  bj-tagex  36988
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