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Theorem snssgOLD 4789
Description: Obsolete version of snssgOLD 4789 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 4647 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21imbi1i 349 . . . 4 ((𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ (𝑥 = 𝐴𝑥𝐵))
32albii 1816 . . 3 (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵))
43a1i 11 . 2 (𝐴𝑉 → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
5 df-ss 3980 . . 3 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
65a1i 11 . 2 (𝐴𝑉 → ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)))
7 clel2g 3659 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ ∀𝑥(𝑥 = 𝐴𝑥𝐵)))
84, 6, 73bitr4rd 312 1 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wcel 2106  wss 3963  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-sn 4632
This theorem is referenced by: (None)
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