Theorem sssn 4787

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 4299	. . . . . . 7 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		ssel 3925	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵}))
3		elsni 4601	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
4	2, 3	syl6 36	. . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵))
5		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
6	4, 5	syl6 36	. . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)))
7	6	ibd 272	. . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴))
8	7	exlimdv 1966	. . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴))
9	1, 8	biimtrid 245	. . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵 ∈ 𝐴))
10		snssi 4746	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
11	9, 10	syl6 36	. . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴))
12	11	anc2li 565	. . . 4 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
13		eqss 3946	. . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
14	12, 13	imbitrrdi 255	. . 3 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵}))
15	14	orrd 877	. 2 ⊢ (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
16		0ss 4350	. . . 4 ⊢ ∅ ⊆ {𝐵}
17		sseq1 3956	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
18	16, 17	mpbiri 261	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
19		eqimss 3989	. . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
20	18, 19	jaoi 871	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
21	15, 20	impbii 212	1 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ⊆ wss 3899  ∅c0 4279  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-dif 3902  df-ss 3916  df-nul 4280  df-sn 4585

This theorem is used by:  eqsn  4790  snsssn  4801  pwsn  4860  frsn  5739  foconst  6811  fin1a2lem12  10489  fpwwe2lem12  10727  gsumval2  18875  0top  23301  minveclem4a  25751  uvtx01vtx  29978  snsssng  33110  pmtrcnelor  33652  0ringsubrg  33812  lvecdim0  34239  locfinref  34473  ordcmp  37235  bj-snmoore  38034  nlpineqsn  38331  uneqsn  45024  mosssn  49924  mosssn2  49926  mofsssn  49955