Theorem snsssn 4372

Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Hypothesis

Ref	Expression
sneqr.1	|- A e. _V

Assertion

Ref	Expression
snsssn	|- ( { A } C_ { B } -> A = B )

Proof of Theorem snsssn

Step	Hyp	Ref	Expression
1		sssn 4358	. 2 |- ( { A } C_ { B } <-> ( { A } = (/) \/ { A } = { B } ) )
2		sneqr.1	. . . . . 6 |- A e. _V
3	2	snnz 4309	. . . . 5 |- { A } =/= (/)
4	3	neii 2796	. . . 4 |- -. { A } = (/)
5	4	pm2.21i 116	. . 3 |- ( { A } = (/) -> A = B )
6	2	sneqr 4371	. . 3 |- ( { A } = { B } -> A = B )
7	5, 6	jaoi 394	. 2 |- ( ( { A } = (/) \/ { A } = { B } ) -> A = B )
8	1, 7	sylbi 207	1 |- ( { A } C_ { B } -> A = B )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {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-3an 1039  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-ne 2795  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178

This theorem is referenced by:  k0004lem3  38447