Theorem snssb 3777

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		ssalel 3189	. 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
2		velsn 3660	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	imbi1i 238	. . 3 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
4	3	albii 1494	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
5		eleq1 2270	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6	5	pm5.74i 180	. . . 4 ⊢ ((𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
7	6	albii 1494	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
8		19.23v 1907	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
9		isset 2783	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
10	9	bicomi 132	. . . 4 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ 𝐴 ∈ V)
11	10	imbi1i 238	. . 3 ⊢ ((∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
12	7, 8, 11	3bitri 206	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
13	1, 4, 12	3bitri 206	1 ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2178  Vcvv 2776   ⊆ wss 3174  {csn 3643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-sn 3649

This theorem is referenced by:  snssg  3778