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Theorem bj-snglex 32661
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 3197 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 pweq 4139 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
32eximi 1759 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4 bj-snglss 32658 . . . . . 6 sngl 𝐴 ⊆ 𝒫 𝐴
5 sseq2 3612 . . . . . 6 (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
64, 5mpbiri 248 . . . . 5 (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
76eximi 1759 . . . 4 (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8 vpwex 4819 . . . . . 6 𝒫 𝑥 ∈ V
98ssex 4772 . . . . 5 (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
109exlimiv 1855 . . . 4 (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
113, 7, 103syl 18 . . 3 (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V)
121, 11sylbi 207 . 2 (𝐴 ∈ V → sngl 𝐴 ∈ V)
13 bj-snglinv 32660 . . 3 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14 bj-snsetex 32651 . . 3 (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
1513, 14syl5eqel 2702 . 2 (sngl 𝐴 ∈ V → 𝐴 ∈ V)
1612, 15impbii 199 1 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wex 1701  wcel 1987  {cab 2607  Vcvv 3190  wss 3560  𝒫 cpw 4136  {csn 4155  sngl bj-csngl 32653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-pw 4138  df-sn 4156  df-pr 4158  df-bj-sngl 32654
This theorem is referenced by:  bj-tagex  32675
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