Theorem bj-snglex 37463

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 3470	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		pweq 4571	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
3	2	eximi 1857	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4		bj-snglss 37460	. . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
5		sseq2 3964	. . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
6	4, 5	mpbiri 260	. . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
7	6	eximi 1857	. . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8		vpwex 5336	. . . . . 6 ⊢ 𝒫 𝑥 ∈ V
9	8	ssex 5279	. . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
10	9	exlimiv 1952	. . . 4 ⊢ (∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
11	3, 7, 10	3syl 18	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 → sngl 𝐴 ∈ V)
12	1, 11	sylbi 219	. 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V)
13		bj-snglinv 37462	. . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14		bj-snsetex 37453	. . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
15	13, 14	eqeltrid 2868	. 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V)
16	12, 15	impbii 211	1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)

Syntax hints:   ↔ wb 208   = wceq 1562  ∃wex 1801   ∈ wcel 2144  {cab 2742  Vcvv 3456   ⊆ wss 3906  𝒫 cpw 4557  {csn 4584  sngl bj-csngl 37455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-pow 5324  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-pw 4559  df-sn 4585  df-pr 4587  df-bj-sngl 37456

This theorem is referenced by:  bj-tagex  37477