Theorem snssOLD 3799

Description: Obsolete version of snss 3808 as of 1-Jan-2025. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
snssOLD.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snssOLD	⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Proof of Theorem snssOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 3686	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	imbi1i 238	. . 3 ⊢ ((𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ (𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
3	2	albii 1518	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
4		ssalel 3215	. 2 ⊢ ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐵))
5		snssOLD.1	. . 3 ⊢ 𝐴 ∈ V
6	5	clel2 2939	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))
7	3, 4, 6	3bitr4ri 213	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ⊆ wss 3200  {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-sn 3675

This theorem is referenced by: (None)