Theorem sssn 2477

Description: The subsets of a singleton.

Assertion

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

Proof of Theorem sssn

Step	Hyp	Ref	Expression
1		ssel 2066	. . . . . . . . . . 11 ⊢ (A ⊆ {B} → (x ∈ A → x ∈ {B}))
2		elsni 2436	. . . . . . . . . . 11 ⊢ (x ∈ {B} → x = B)
3	1, 2	syl6 22	. . . . . . . . . 10 ⊢ (A ⊆ {B} → (x ∈ A → x = B))
4		eleq1 1537	. . . . . . . . . 10 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
5	3, 4	syl6 22	. . . . . . . . 9 ⊢ (A ⊆ {B} → (x ∈ A → (x ∈ A ↔ B ∈ A)))
6	5	ibd 596	. . . . . . . 8 ⊢ (A ⊆ {B} → (x ∈ A → B ∈ A))
7	6	19.23adv 1216	. . . . . . 7 ⊢ (A ⊆ {B} → (∃x x ∈ A → B ∈ A))
8		n0 2293	. . . . . . 7 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
9	7, 8	syl5ib 206	. . . . . 6 ⊢ (A ⊆ {B} → (¬ A = ∅ → B ∈ A))
10		snssi 2470	. . . . . 6 ⊢ (B ∈ A → {B} ⊆ A)
11	9, 10	syl6 22	. . . . 5 ⊢ (A ⊆ {B} → (¬ A = ∅ → {B} ⊆ A))
12	11	anc2li 302	. . . 4 ⊢ (A ⊆ {B} → (¬ A = ∅ → (A ⊆ {B} ⋀ {B} ⊆ A)))
13		eqss 2080	. . . 4 ⊢ (A = {B} ↔ (A ⊆ {B} ⋀ {B} ⊆ A))
14	12, 13	syl6ibr 213	. . 3 ⊢ (A ⊆ {B} → (¬ A = ∅ → A = {B}))
15	14	orrd 233	. 2 ⊢ (A ⊆ {B} → (A = ∅ ⋁ A = {B}))
16		0ss 2305	. . . 4 ⊢ ∅ ⊆ {B}
17		sseq1 2085	. . . 4 ⊢ (A = ∅ → (A ⊆ {B} ↔ ∅ ⊆ {B}))
18	16, 17	mpbiri 194	. . 3 ⊢ (A = ∅ → A ⊆ {B})
19		eqimss 2112	. . 3 ⊢ (A = {B} → A ⊆ {B})
20	18, 19	jaoi 341	. 2 ⊢ ((A = ∅ ⋁ A = {B}) → A ⊆ {B})
21	15, 20	impbi 157	1 ⊢ (A ⊆ {B} ↔ (A = ∅ ⋁ A = {B}))

Syntax hints:  ¬ wn 2   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wex 982   ⊆ wss 2050  ∅c0 2283  {csn 2413

This theorem is referenced by:  eqsn 2478  sspr 2479  snsssn 2482  pwsn 2504  foconst 3689  0top 7634  sn0top 7644  top2usne 10535

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-sn 2416  df-pr 2417

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