Theorem bj-snglex 33800

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 3428	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		pweq 4425	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
3	2	eximi 1797	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4		bj-snglss 33797	. . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
5		sseq2 3884	. . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
6	4, 5	mpbiri 250	. . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
7	6	eximi 1797	. . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8		vpwex 5131	. . . . . 6 ⊢ 𝒫 𝑥 ∈ V
9	8	ssex 5081	. . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
10	9	exlimiv 1889	. . . 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 209	. 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V)
13		bj-snglinv 33799	. . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14		bj-snsetex 33790	. . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
15	13, 14	syl5eqel 2871	. 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V)
16	12, 15	impbii 201	1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)

Syntax hints:   ↔ wb 198   = wceq 1507  ∃wex 1742   ∈ wcel 2050  {cab 2759  Vcvv 3416   ⊆ wss 3830  𝒫 cpw 4422  {csn 4441  sngl bj-csngl 33792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-rex 3095  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-pw 4424  df-sn 4442  df-pr 4444  df-bj-sngl 33793

This theorem is referenced by:  bj-tagex  33814