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Theorem snssiALTVD 44825
Description: Virtual deduction proof of snssiALT 44826. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snssiALTVD (𝐴𝐵 → {𝐴} ⊆ 𝐵)

Proof of Theorem snssiALTVD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-ss 3980 . . 3 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
2 idn1 44572 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn2 44611 . . . . . . 7 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4 velsn 4647 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
53, 4e2bi 44630 . . . . . 6 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6 eleq1a 2834 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
72, 5, 6e12 44722 . . . . 5 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥𝐵   )
87in2 44603 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥𝐵)   )
98gen11 44614 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)   )
10 biimpr 220 . . 3 (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) → {𝐴} ⊆ 𝐵))
111, 9, 10e01 44689 . 2 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
1211in1 44569 1 (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  df-vd1 44568  df-vd2 44576
This theorem is referenced by: (None)
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