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Theorem sssn 4490
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.
StepHypRef Expression
1 neq0 4075 . . . . . . 7 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
2 ssel 3744 . . . . . . . . . . 11 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
3 elsni 4331 . . . . . . . . . . 11 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
42, 3syl6 35 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
5 eleq1 2837 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
64, 5syl6 35 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
76ibd 258 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
87exlimdv 2012 . . . . . . 7 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
91, 8syl5bi 232 . . . . . 6 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵𝐴))
10 snssi 4472 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
119, 10syl6 35 . . . . 5 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴))
1211anc2li 537 . . . 4 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
13 eqss 3765 . . . 4 (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
1412, 13syl6ibr 242 . . 3 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵}))
1514orrd 843 . 2 (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
16 0ss 4114 . . . 4 ∅ ⊆ {𝐵}
17 sseq1 3773 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
1816, 17mpbiri 248 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
19 eqimss 3804 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
2018, 19jaoi 837 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
2115, 20impbii 199 1 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 826   = wceq 1630  wex 1851  wcel 2144  wss 3721  c0 4061  {csn 4314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-in 3728  df-ss 3735  df-nul 4062  df-sn 4315
This theorem is referenced by:  eqsn  4493  snsssn  4503  pwsn  4564  frsn  5329  foconst  6267  fin1a2lem12  9434  fpwwe2lem13  9665  gsumval2  17487  0top  21007  minveclem4a  23419  uvtx01vtx  26524  uvtxa01vtx0OLD  26526  locfinref  30242  ordcmp  32777  bj-snmoore  33393  uneqsn  38840
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