Theorem bj-snglex 37637

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 3468	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		pweq 4575	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
3	2	eximi 1864	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝒫 𝑥 = 𝒫 𝐴)
4		bj-snglss 37634	. . . . . 6 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
5		sseq2 3962	. . . . . 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 1864	. . . 4 ⊢ (∃𝑥𝒫 𝑥 = 𝒫 𝐴 → ∃𝑥sngl 𝐴 ⊆ 𝒫 𝑥)
8		vpwex 5347	. . . . . 6 ⊢ 𝒫 𝑥 ∈ V
9	8	ssex 5290	. . . . 5 ⊢ (sngl 𝐴 ⊆ 𝒫 𝑥 → sngl 𝐴 ∈ V)
10	9	exlimiv 1959	. . . 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 37636	. . 3 ⊢ 𝐴 = {𝑦 ∣ {𝑦} ∈ sngl 𝐴}
14		bj-snsetex 37627	. . 3 ⊢ (sngl 𝐴 ∈ V → {𝑦 ∣ {𝑦} ∈ sngl 𝐴} ∈ V)
15	13, 14	eqeltrid 2866	. 2 ⊢ (sngl 𝐴 ∈ V → 𝐴 ∈ V)
16	12, 15	impbii 212	1 ⊢ (𝐴 ∈ V ↔ sngl 𝐴 ∈ V)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4561  {csn 4588  sngl bj-csngl 37629

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-pw 4563  df-sn 4589  df-pr 4591  df-bj-sngl 37630

This theorem is used by:  bj-tagex  37651