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Theorem snsslVD 39581
 Description: Virtual deduction proof of snssl 39582. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
snsslVD.1 𝐴 ∈ V
Assertion
Ref Expression
snsslVD ({𝐴} ⊆ 𝐵𝐴𝐵)

Proof of Theorem snsslVD
StepHypRef Expression
1 idn1 39310 . . 3 (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2 snsslVD.1 . . . 4 𝐴 ∈ V
32snid 4353 . . 3 𝐴 ∈ {𝐴}
4 ssel2 3739 . . 3 (({𝐴} ⊆ 𝐵𝐴 ∈ {𝐴}) → 𝐴𝐵)
51, 3, 4e10an 39440 . 2 (   {𝐴} ⊆ 𝐵   ▶   𝐴𝐵   )
65in1 39307 1 ({𝐴} ⊆ 𝐵𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2139  Vcvv 3340   ⊆ wss 3715  {csn 4321 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-sn 4322  df-vd1 39306 This theorem is referenced by: (None)
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