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Theorem snsslVD 41318
Description: Virtual deduction proof of snssl 41319. (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 41063 . . 3 (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2 snsslVD.1 . . . 4 𝐴 ∈ V
32snid 4577 . . 3 𝐴 ∈ {𝐴}
4 ssel2 3941 . . 3 (({𝐴} ⊆ 𝐵𝐴 ∈ {𝐴}) → 𝐴𝐵)
51, 3, 4e10an 41184 . 2 (   {𝐴} ⊆ 𝐵   ▶   𝐴𝐵   )
65in1 41060 1 ({𝐴} ⊆ 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  Vcvv 3473  wss 3913  {csn 4543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-v 3475  df-in 3920  df-ss 3930  df-sn 4544  df-vd1 41059
This theorem is referenced by: (None)
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