Theorem snssb 3755

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		dfss2 3172	. 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
2		velsn 3639	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
3	2	imbi1i 238	. . 3 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
4	3	albii 1484	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
5		eleq1 2259	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
6	5	pm5.74i 180	. . . 4 ⊢ ((𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
7	6	albii 1484	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
8		19.23v 1897	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝐴 ∈ 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵))
9		isset 2769	. . . . 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 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  {csn 3622

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-sn 3628

This theorem is referenced by:  snssg  3756