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Theorem sssn 4719
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 4259 . . . . . . 7 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
2 ssel 3908 . . . . . . . . . . 11 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
3 elsni 4542 . . . . . . . . . . 11 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
42, 3syl6 35 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
5 eleq1 2877 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
64, 5syl6 35 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
76ibd 272 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
87exlimdv 1934 . . . . . . 7 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
91, 8syl5bi 245 . . . . . 6 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵𝐴))
10 snssi 4701 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
119, 10syl6 35 . . . . 5 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴))
1211anc2li 559 . . . 4 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
13 eqss 3930 . . . 4 (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
1412, 13syl6ibr 255 . . 3 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵}))
1514orrd 860 . 2 (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
16 0ss 4304 . . . 4 ∅ ⊆ {𝐵}
17 sseq1 3940 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
1816, 17mpbiri 261 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
19 eqimss 3971 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
2018, 19jaoi 854 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
2115, 20impbii 212 1 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2111  wss 3881  c0 4243  {csn 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526
This theorem is referenced by:  eqsn  4722  snsssn  4732  pwsn  4792  frsn  5603  foconst  6578  fin1a2lem12  9822  fpwwe2lem13  10053  gsumval2  17888  0top  21588  minveclem4a  24034  uvtx01vtx  27187  snsssng  30284  pmtrcnelor  30785  lvecdim0  31093  locfinref  31194  ordcmp  33908  bj-snmoore  34528  nlpineqsn  34825  uneqsn  40724
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