Theorem ss1o0el1 4212

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 2768	. . . 4 |- ( (/) e. A -> E. x x e. A )
2		sssnm 3769	. . . 4 |- ( E. x x e. A -> ( A C_ { (/) } <-> A = { (/) } ) )
3	1, 2	syl 14	. . 3 |- ( (/) e. A -> ( A C_ { (/) } <-> A = { (/) } ) )
4	3	biimpcd 159	. 2 |- ( A C_ { (/) } -> ( (/) e. A -> A = { (/) } ) )
5		0ex 4145	. . . 4 |- (/) e. _V
6	5	snid 3638	. . 3 |- (/) e. { (/) }
7		eleq2 2253	. . 3 |- ( A = { (/) } -> ( (/) e. A <-> (/) e. { (/) } ) )
8	6, 7	mpbiri 168	. 2 |- ( A = { (/) } -> (/) e. A )
9	4, 8	impbid1 142	1 |- ( A C_ { (/) } -> ( (/) e. A <-> A = { (/) } ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   C_ wss 3144   (/)c0 3437   {csn 3607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4144

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613

This theorem is referenced by:  exmid01  4213  ss1o0el1o  6930