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Theorem snssl 39564
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 4460. The proof of this theorem was automatically generated from snsslVD 39563 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 4353 . 2 𝐴 ∈ {𝐴}
3 ssel2 3739 . 2 (({𝐴} ⊆ 𝐵𝐴 ∈ {𝐴}) → 𝐴𝐵)
42, 3mpan2 709 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
This theorem is referenced by: (None)
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