Theorem sssn 4298

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 3889	. . . . . . 7 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		ssel 3562	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝐵}))
3		elsni 4142	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
4	2, 3	syl6 34	. . . . . . . . . 10 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝑥 = 𝐵))
5		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴))
6	4, 5	syl6 34	. . . . . . . . 9 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐵 ∈ 𝐴)))
7	6	ibd 257	. . . . . . . 8 ⊢ (𝐴 ⊆ {𝐵} → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴))
8	7	exlimdv 1848	. . . . . . 7 ⊢ (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐴))
9	1, 8	syl5bi 231	. . . . . 6 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵 ∈ 𝐴))
10		snssi 4280	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
11	9, 10	syl6 34	. . . . 5 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴))
12	11	anc2li 578	. . . 4 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
13		eqss 3583	. . . 4 ⊢ (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
14	12, 13	syl6ibr 241	. . 3 ⊢ (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵}))
15	14	orrd 392	. 2 ⊢ (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
16		0ss 3924	. . . 4 ⊢ ∅ ⊆ {𝐵}
17		sseq1 3589	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
18	16, 17	mpbiri 247	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
19		eqimss 3620	. . 3 ⊢ (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
20	18, 19	jaoi 393	. 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
21	15, 20	impbii 198	1 ⊢ (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126

This theorem is referenced by:  eqsn  4301  eqsnOLD  4302  snsssn  4312  pwsn  4366  frsn  5112  foconst  6039  fin1a2lem12  9116  fpwwe2lem13  9343  gsumval2  17103  0top  20598  minveclem4a  23009  uvtx01vtx  26020  locfinref  29236  ordcmp  31616  uneqsn  37341  uvtxa01vtx0  40623