Theorem bj-snglex 36968

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 3464	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		pweq 4580	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
3	2	eximi 1835	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4		bj-snglss 36965	. . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
5		sseq2 3976	. . . . . 6 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → (sngl 𝐴 ⊆ 𝒫 𝑥 ↔ sngl 𝐴 ⊆ 𝒫 𝐴))
6	4, 5	mpbiri 258	. . . . 5 ⊢ (𝒫 𝑥 = 𝒫 𝐴 → sngl 𝐴 ⊆ 𝒫 𝑥)
7	6	eximi 1835	. . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8		vpwex 5335	. . . . . 6 ⊢ 𝒫 𝑥 ∈ V
9	8	ssex 5279	. . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
10	9	exlimiv 1930	. . . 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 217	. 2 ⊢ (𝐴 ∈ V → sngl 𝐴 ∈ V)
13		bj-snglinv 36967	. . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14		bj-snsetex 36958	. . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
15	13, 14	eqeltrid 2833	. 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V)
16	12, 15	impbii 209	1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)

Syntax hints:   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  Vcvv 3450   ⊆ wss 3917  𝒫 cpw 4566  {csn 4592  sngl bj-csngl 36960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-pw 4568  df-sn 4593  df-pr 4595  df-bj-sngl 36961

This theorem is referenced by:  bj-tagex  36982