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Theorem snssl 41523
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 4682. The proof of this theorem was automatically generated from snsslVD 41522 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 4564 . 2 𝐴 ∈ {𝐴}
3 ssel2 3913 . 2 (({𝐴} ⊆ 𝐵𝐴 ∈ {𝐴}) → 𝐴𝐵)
42, 3mpan2 690 1 ({𝐴} ⊆ 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  Vcvv 3444  wss 3884  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-in 3891  df-ss 3901  df-sn 4529
This theorem is referenced by: (None)
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