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Theorem snssgOLD 4744
Description: Obsolete version of snssgOLD 4744 as of 1-Jan-2025. (Contributed by NM, 22-Jul-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snssgOLD (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))

Proof of Theorem snssgOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 velsn 4601 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21imbi1i 349 . . . 4 ((𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ (𝑥 = 𝐴𝑥𝐵))
32albii 1821 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
43a1i 11 . 2 (𝐴𝑉 → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
5 dfss2 3929 . . 3 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
65a1i 11 . 2 (𝐴𝑉 → ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)))
7 clel2g 3608 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
84, 6, 73bitr4rd 311 1 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  wcel 2106  wss 3909  {csn 4585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3446  df-in 3916  df-ss 3926  df-sn 4586
This theorem is referenced by: (None)
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