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

Theorem snssl 44263
Description: If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 4785. The proof of this theorem was automatically generated from snsslVD 44262 using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
snssl.1 𝐴 ∈ V
Assertion
Ref Expression
snssl ({𝐴} ⊆ 𝐵𝐴𝐵)

Proof of Theorem snssl
StepHypRef Expression
1 snssl.1 . . 3 𝐴 ∈ V
21snid 4660 . 2 𝐴 ∈ {𝐴}
3 ssel2 3973 . 2 (({𝐴} ⊆ 𝐵𝐴 ∈ {𝐴}) → 𝐴𝐵)
42, 3mpan2 690 1 ({𝐴} ⊆ 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3470  wss 3945  {csn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-ss 3962  df-sn 4625
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator