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Theorem snssiALTVD 40586
 Description: Virtual deduction proof of snssiALT 40587. (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 3846 . . 3 ({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵))
2 idn1 40333 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn2 40372 . . . . . . 7 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 ∈ {𝐴}   )
4 velsn 4457 . . . . . . 7 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
53, 4e2bi 40391 . . . . . 6 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥 = 𝐴   )
6 eleq1a 2861 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
72, 5, 6e12 40483 . . . . 5 (   𝐴𝐵   ,   𝑥 ∈ {𝐴}   ▶   𝑥𝐵   )
87in2 40364 . . . 4 (   𝐴𝐵   ▶   (𝑥 ∈ {𝐴} → 𝑥𝐵)   )
98gen11 40375 . . 3 (   𝐴𝐵   ▶   𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)   )
10 biimpr 212 . . 3 (({𝐴} ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵)) → (∀𝑥(𝑥 ∈ {𝐴} → 𝑥𝐵) → {𝐴} ⊆ 𝐵))
111, 9, 10e01 40450 . 2 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
1211in1 40330 1 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505   = wceq 1507   ∈ wcel 2050   ⊆ wss 3829  {csn 4441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-in 3836  df-ss 3843  df-sn 4442  df-vd1 40329  df-vd2 40337 This theorem is referenced by: (None)
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