Theorem snssl 39065

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 4316. The proof of this theorem was automatically generated from snsslVD 39064 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 e. _V

Assertion

Ref	Expression
snssl	|- ( { A } C_ B -> A e. B )

Proof of Theorem snssl

Step	Hyp	Ref	Expression
1		snssl.1	. . 3 |- A e. _V
2	1	snid 4208	. 2 |- A e. { A }
3		ssel2 3598	. 2 |- ( ( { A } C_ B /\ A e. { A } ) -> A e. B )
4	2, 3	mpan2 707	1 |- ( { A } C_ B -> A e. B )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-sn 4178

This theorem is referenced by: (None)