Theorem snssgOLD 3849

Description: Obsolete version of snssgOLD 3849 as of 1-Jan-2025. (Contributed by NM, 22-Jul-2001.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem snssgOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2301	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2		sneq 3719	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	2	sseq1d 3277	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4		vex 2824	. . 3 ⊢ 𝑥 ∈ V
5	4	snss 3848	. 2 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵)
6	1, 3, 5	vtoclbg 2884	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  {csn 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-sn 3714

This theorem is referenced by: (None)