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Theorem snsslVD 41040
Description: Virtual deduction proof of snssl 41041. (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 40785 . . 3 (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2 snsslVD.1 . . . 4 𝐴 ∈ V
32snid 4591 . . 3 𝐴 ∈ {𝐴}
4 ssel2 3959 . . 3 (({𝐴} ⊆ 𝐵𝐴 ∈ {𝐴}) → 𝐴𝐵)
51, 3, 4e10an 40906 . 2 (   {𝐴} ⊆ 𝐵   ▶   𝐴𝐵   )
65in1 40782 1 ({𝐴} ⊆ 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492  wss 3933  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-ss 3949  df-sn 4558  df-vd1 40781
This theorem is referenced by: (None)
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