Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  snsslVD Structured version   Visualization version   GIF version

Theorem snsslVD 44271
Description: Virtual deduction proof of snssl 44272. (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 44016 . . 3 (   {𝐴} ⊆ 𝐵   ▶   {𝐴} ⊆ 𝐵   )
2 snsslVD.1 . . . 4 𝐴 ∈ V
32snid 4667 . . 3 𝐴 ∈ {𝐴}
4 ssel2 3975 . . 3 (({𝐴} ⊆ 𝐵𝐴 ∈ {𝐴}) → 𝐴𝐵)
51, 3, 4e10an 44137 . 2 (   {𝐴} ⊆ 𝐵   ▶   𝐴𝐵   )
65in1 44013 1 ({𝐴} ⊆ 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3471  wss 3947  {csn 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3473  df-in 3954  df-ss 3964  df-sn 4631  df-vd1 44012
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator