Theorem snssiALT 39063

Description: If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4339. This theorem was automatically generated from snssiALTVD 39062 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem snssiALT

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 4193	. . . 4 |- ( x e. { A } <-> x = A )
2		eleq1a 2696	. . . 4 |- ( A e. B -> ( x = A -> x e. B ) )
3	1, 2	syl5bi 232	. . 3 |- ( A e. B -> ( x e. { A } -> x e. B ) )
4	3	alrimiv 1855	. 2 |- ( A e. B -> A. x ( x e. { A } -> x e. B ) )
5		dfss2 3591	. 2 |- ( { A } C_ B <-> A. x ( x e. { A } -> x e. B ) )
6	4, 5	sylibr 224	1 |- ( A e. B -> { A } C_ B )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990   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)