Theorem sssn 3864

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

Assertion

Ref	Expression
sssn	⊢ (A ⊆ {B} ↔ (A = ∅ ∨ A = {B}))

Proof of Theorem sssn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neq0 3560	. . . . . . 7 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
2		ssel 3267	. . . . . . . . . . 11 ⊢ (A ⊆ {B} → (x ∈ A → x ∈ {B}))
3		elsni 3757	. . . . . . . . . . 11 ⊢ (x ∈ {B} → x = B)
4	2, 3	syl6 29	. . . . . . . . . 10 ⊢ (A ⊆ {B} → (x ∈ A → x = B))
5		eleq1 2413	. . . . . . . . . 10 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
6	4, 5	syl6 29	. . . . . . . . 9 ⊢ (A ⊆ {B} → (x ∈ A → (x ∈ A ↔ B ∈ A)))
7	6	ibd 234	. . . . . . . 8 ⊢ (A ⊆ {B} → (x ∈ A → B ∈ A))
8	7	exlimdv 1636	. . . . . . 7 ⊢ (A ⊆ {B} → (∃x x ∈ A → B ∈ A))
9	1, 8	syl5bi 208	. . . . . 6 ⊢ (A ⊆ {B} → (¬ A = ∅ → B ∈ A))
10		snssi 3852	. . . . . 6 ⊢ (B ∈ A → {B} ⊆ A)
11	9, 10	syl6 29	. . . . 5 ⊢ (A ⊆ {B} → (¬ A = ∅ → {B} ⊆ A))
12	11	anc2li 540	. . . 4 ⊢ (A ⊆ {B} → (¬ A = ∅ → (A ⊆ {B} ∧ {B} ⊆ A)))
13		eqss 3287	. . . 4 ⊢ (A = {B} ↔ (A ⊆ {B} ∧ {B} ⊆ A))
14	12, 13	syl6ibr 218	. . 3 ⊢ (A ⊆ {B} → (¬ A = ∅ → A = {B}))
15	14	orrd 367	. 2 ⊢ (A ⊆ {B} → (A = ∅ ∨ A = {B}))
16		0ss 3579	. . . 4 ⊢ ∅ ⊆ {B}
17		sseq1 3292	. . . 4 ⊢ (A = ∅ → (A ⊆ {B} ↔ ∅ ⊆ {B}))
18	16, 17	mpbiri 224	. . 3 ⊢ (A = ∅ → A ⊆ {B})
19		eqimss 3323	. . 3 ⊢ (A = {B} → A ⊆ {B})
20	18, 19	jaoi 368	. 2 ⊢ ((A = ∅ ∨ A = {B}) → A ⊆ {B})
21	15, 20	impbii 180	1 ⊢ (A ⊆ {B} ↔ (A = ∅ ∨ A = {B}))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ∅c0 3550  {csn 3737

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741

This theorem is referenced by:  eqsn  3867  snsssn  3873  pwsn  3881  unsneqsn  3887  foconst  5280