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Theorem sssn 4830
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 4346 . . . . . . 7 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
2 ssel 3976 . . . . . . . . . . 11 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
3 elsni 4646 . . . . . . . . . . 11 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
42, 3syl6 35 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
5 eleq1 2822 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
64, 5syl6 35 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
76ibd 269 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
87exlimdv 1937 . . . . . . 7 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
91, 8biimtrid 241 . . . . . 6 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐵𝐴))
10 snssi 4812 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
119, 10syl6 35 . . . . 5 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → {𝐵} ⊆ 𝐴))
1211anc2li 557 . . . 4 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
13 eqss 3998 . . . 4 (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
1412, 13imbitrrdi 251 . . 3 (𝐴 ⊆ {𝐵} → (¬ 𝐴 = ∅ → 𝐴 = {𝐵}))
1514orrd 862 . 2 (𝐴 ⊆ {𝐵} → (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
16 0ss 4397 . . . 4 ∅ ⊆ {𝐵}
17 sseq1 4008 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵} ↔ ∅ ⊆ {𝐵}))
1816, 17mpbiri 258 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵})
19 eqimss 4041 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
2018, 19jaoi 856 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵})
2115, 20impbii 208 1 (𝐴 ⊆ {𝐵} ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wss 3949  c0 4323  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-sn 4630
This theorem is referenced by:  eqsn  4833  snsssn  4843  pwsn  4901  frsn  5764  foconst  6821  fin1a2lem12  10406  fpwwe2lem12  10637  gsumval2  18605  0top  22486  minveclem4a  24947  uvtx01vtx  28654  snsssng  31752  pmtrcnelor  32252  0ringsubrg  32379  lvecdim0  32691  locfinref  32821  ordcmp  35332  bj-snmoore  35994  nlpineqsn  36289  uneqsn  42776  mosssn  47499  mosssn2  47501  mofsssn  47512
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