Theorem ss1o0el1 4257

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 2793	. . . 4 |- ( (/) e. A -> E. x x e. A )
2		sssnm 3808	. . . 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 4187	. . . 4 |- (/) e. _V
6	5	snid 3674	. . 3 |- (/) e. { (/) }
7		eleq2 2271	. . 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 1373   E.wex 1516   e. wcel 2178   C_ wss 3174   (/)c0 3468   {csn 3643

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-nul 4186

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469  df-sn 3649

This theorem is referenced by:  exmid01  4258  ss1o0el1o  7036