Theorem sssn 3865

Description: The subsets of a singleton. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
sssn	|- (A C_ {B} <-> (A = (/) \/ A = {B}))

Proof of Theorem sssn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 3561	. . . . . . 7 |- (-. A = (/) <-> E.x x e. A)
2		ssel 3268	. . . . . . . . . . 11 |- (A C_ {B} -> (x e. A -> x e. {B}))
3		elsni 3758	. . . . . . . . . . 11 |- (x e. {B} -> x = B)
4	2, 3	syl6 29	. . . . . . . . . 10 |- (A C_ {B} -> (x e. A -> x = B))
5		eleq1 2413	. . . . . . . . . 10 |- (x = B -> (x e. A <-> B e. A))
6	4, 5	syl6 29	. . . . . . . . 9 |- (A C_ {B} -> (x e. A -> (x e. A <-> B e. A)))
7	6	ibd 234	. . . . . . . 8 |- (A C_ {B} -> (x e. A -> B e. A))
8	7	exlimdv 1636	. . . . . . 7 |- (A C_ {B} -> (E.x x e. A -> B e. A))
9	1, 8	syl5bi 208	. . . . . 6 |- (A C_ {B} -> (-. A = (/) -> B e. A))
10		snssi 3853	. . . . . 6 |- (B e. A -> {B} C_ A)
11	9, 10	syl6 29	. . . . 5 |- (A C_ {B} -> (-. A = (/) -> {B} C_ A))
12	11	anc2li 540	. . . 4 |- (A C_ {B} -> (-. A = (/) -> (A C_ {B} /\ {B} C_ A)))
13		eqss 3288	. . . 4 |- (A = {B} <-> (A C_ {B} /\ {B} C_ A))
14	12, 13	syl6ibr 218	. . 3 |- (A C_ {B} -> (-. A = (/) -> A = {B}))
15	14	orrd 367	. 2 |- (A C_ {B} -> (A = (/) \/ A = {B}))
16		0ss 3580	. . . 4 |- (/) C_ {B}
17		sseq1 3293	. . . 4 |- (A = (/) -> (A C_ {B} <-> (/) C_ {B}))
18	16, 17	mpbiri 224	. . 3 |- (A = (/) -> A C_ {B})
19		eqimss 3324	. . 3 |- (A = {B} -> A C_ {B})
20	18, 19	jaoi 368	. 2 |- ((A = (/) \/ A = {B}) -> A C_ {B})
21	15, 20	impbii 180	1 |- (A C_ {B} <-> (A = (/) \/ A = {B}))

Syntax hints:  -. wn 3   <-> wb 176   \/ wo 357   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710   C_ wss 3258  (/)c0 3551  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742

This theorem is referenced by:  eqsn  3868  snsssn  3874  pwsn  3882  unsneqsn  3888  foconst  5281