Theorem ss1o0el1 4176

Description: A subclass of { (/) } contains the empty set if and only if it equals { (/) }. (Contributed by BJ and Jim Kingdon, 9-Aug-2024.)

Assertion

Ref	Expression
ss1o0el1	|- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )

Proof of Theorem ss1o0el1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2742	. . . 4 |- ( (/) e. A -> E. x x e. A )
2		sssnm 3734	. . . 4 |- ( E. x x e. A -> ( A C_ { (/) } <-> A = { (/) } ) )
3	1, 2	syl 14	. . 3 |- ( (/) e. A -> ( A C_ { (/) } <-> A = { (/) } ) )
4	3	biimpcd 158	. 2 |- ( A C_ { (/) } -> ( (/) e. A -> A = { (/) } ) )
5		0ex 4109	. . . 4 |- (/) e. _V
6	5	snid 3607	. . 3 |- (/) e. { (/) }
7		eleq2 2230	. . 3 |- ( A = { (/) } -> ( (/) e. A <-> (/) e. { (/) } ) )
8	6, 7	mpbiri 167	. 2 |- ( A = { (/) } -> (/) e. A )
9	4, 8	impbid1 141	1 |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   C_ wss 3116   (/)c0 3409   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-nul 4108

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582

This theorem is referenced by:  exmid01  4177  ss1o0el1o  6878