Theorem ss1o0el1 4158

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 2728	. . . 4 |- ( (/) e. A -> E. x x e. A )
2		sssnm 3717	. . . 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 4091	. . . 4 |- (/) e. _V
6	5	snid 3591	. . 3 |- (/) e. { (/) }
7		eleq2 2221	. . 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 1335   E.wex 1472   e. wcel 2128   C_ wss 3102   (/)c0 3394   {csn 3560

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566

This theorem is referenced by:  exmid01  4159  ss1o0el1o  6854