Theorem sssn 4781

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 4298	. . . . . . 7 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		ssel 3903	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵}))
3		elsni 4597	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
4	2, 3	syl6 36	. . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵))
5		eleq1 2848	. . . . . . . . . 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 1965	. . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴))
9	1, 8	biimtrid 245	. . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵 ∈ 𝐴))
10		snssi 4742	. . . . . 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 3924	. . . 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 4349	. . . 4 ⊢ ∅ ⊆ {𝐵}
17		sseq1 3934	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
18	16, 17	mpbiri 261	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
19		eqimss 3967	. . 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 1569  ∃wex 1811   ∈ wcel 2144   ⊆ wss 3896  ∅c0 4277  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-dif 4046  df-nul 4278  df-sn 4585

This theorem is used by:  eqsn  4784  snsssn  4795  pwsn  4826  frsn  5740  foconst  6816  fin1a2lem12  10496  fpwwe2lem12  10734  gsumval2  18884  0top  23310  minveclem4a  25760  uvtx01vtx  29987  snsssng  33161  pmtrcnelor  33702  0ringsubrg  33862  lvecdim0  34289  locfinref  34523  0elscott  35813  ordcmp  37359  bj-snmoore  38170  nlpineqsn  38468  uneqsn  45162  mosssn  50062  mosssn2  50064  mofsssn  50093