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Theorem bj-snglex 34370
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 3481 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 pweq 4527 . . . . 5 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
32eximi 1836 . . . 4 (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4 bj-snglss 34367 . . . . . 6 sngl 𝐴 ⊆ 𝒫 𝐴
5 sseq2 3968 . . . . . 6 (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
64, 5mpbiri 261 . . . . 5 (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
76eximi 1836 . . . 4 (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8 vpwex 5255 . . . . . 6 𝒫 𝑥 ∈ V
98ssex 5201 . . . . 5 (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
109exlimiv 1931 . . . 4 (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
113, 7, 103syl 18 . . 3 (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V)
121, 11sylbi 220 . 2 (𝐴 ∈ V → sngl 𝐴 ∈ V)
13 bj-snglinv 34369 . . 3 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14 bj-snsetex 34360 . . 3 (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
1513, 14eqeltrid 2918 . 2 (sngl 𝐴 ∈ V → 𝐴 ∈ V)
1612, 15impbii 212 1 (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wex 1781  wcel 2114  {cab 2800  Vcvv 3469  wss 3908  𝒫 cpw 4511  {csn 4539  sngl bj-csngl 34362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-pw 4513  df-sn 4540  df-pr 4542  df-bj-sngl 34363
This theorem is referenced by:  bj-tagex  34384
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