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Theorem snssiALTVD 38084
 Description: Virtual deduction proof of snssiALT 38085. (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 dfss2 3557 . . 3 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
2 idn1 37811 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn2 37859 . . . . . . 7 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4 velsn 4141 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
53, 4e2bi 37878 . . . . . 6 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6 eleq1a 2683 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
72, 5, 6e12 37972 . . . . 5 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥𝐵   )
87in2 37851 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥𝐵)   )
98gen11 37862 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)   )
10 biimpr 209 . . 3 (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) → {𝐴} ⊆ 𝐵))
111, 9, 10e01 37937 . 2 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
1211in1 37808 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-sn 4126  df-vd1 37807  df-vd2 37815 This theorem is referenced by: (None)
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