Theorem bj-snglex 37497

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.

Step	Hyp	Ref	Expression
1		isset 3477	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		pweq 4581	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
3	2	eximi 1862	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4		bj-snglss 37494	. . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
5		sseq2 3971	. . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
6	4, 5	mpbiri 261	. . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
7	6	eximi 1862	. . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8		vpwex 5349	. . . . . 6 ⊢ 𝒫 𝑥 ∈ V
9	8	ssex 5292	. . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
10	9	exlimiv 1957	. . . 4 ⊢ (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
11	3, 7, 10	3syl 19	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V)
12	1, 11	sylbi 220	. 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V)
13		bj-snglinv 37496	. . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14		bj-snsetex 37487	. . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
15	13, 14	eqeltrid 2873	. 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V)
16	12, 15	impbii 212	1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)

Syntax hints:   ↔ wb 209   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  Vcvv 3463   ⊆ wss 3913  𝒫 cpw 4567  {csn 4594  sngl bj-csngl 37489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-pw 4569  df-sn 4595  df-pr 4597  df-bj-sngl 37490

This theorem is referenced by:  bj-tagex  37511