Theorem snssb 4764

Description: Characterization of the inclusion of a singleton in a class. (Contributed by BJ, 1-Jan-2025.)

Assertion

Ref	Expression
snssb	⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))

Proof of Theorem snssb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3950	. 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
2		velsn 4624	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	imbi1i 349	. . 3 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
4	3	albii 1818	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
5		eleq1 2821	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6	5	pm5.74i 271	. . . 4 ⊢ ((𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
7	6	albii 1818	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
8		19.23v 1941	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
9		isset 3478	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	9	bicomi 224	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ V)
11	10	imbi1i 349	. . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
12	7, 8, 11	3bitri 297	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
13	1, 4, 12	3bitri 297	1 ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3464   ⊆ wss 3933  {csn 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3466  df-ss 3950  df-sn 4609

This theorem is referenced by:  snssg  4765