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Theorem sssn 3864
 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 3560 . . . . . . 7 A = x x A)
2 ssel 3267 . . . . . . . . . . 11 (A {B} → (x Ax {B}))
3 elsni 3757 . . . . . . . . . . 11 (x {B} → x = B)
42, 3syl6 29 . . . . . . . . . 10 (A {B} → (x Ax = B))
5 eleq1 2413 . . . . . . . . . 10 (x = B → (x AB A))
64, 5syl6 29 . . . . . . . . 9 (A {B} → (x A → (x AB A)))
76ibd 234 . . . . . . . 8 (A {B} → (x AB A))
87exlimdv 1636 . . . . . . 7 (A {B} → (x x AB A))
91, 8syl5bi 208 . . . . . 6 (A {B} → (¬ A = B A))
10 snssi 3852 . . . . . 6 (B A → {B} A)
119, 10syl6 29 . . . . 5 (A {B} → (¬ A = → {B} A))
1211anc2li 540 . . . 4 (A {B} → (¬ A = → (A {B} {B} A)))
13 eqss 3287 . . . 4 (A = {B} ↔ (A {B} {B} A))
1412, 13syl6ibr 218 . . 3 (A {B} → (¬ A = A = {B}))
1514orrd 367 . 2 (A {B} → (A = A = {B}))
16 0ss 3579 . . . 4 {B}
17 sseq1 3292 . . . 4 (A = → (A {B} ↔ {B}))
1816, 17mpbiri 224 . . 3 (A = A {B})
19 eqimss 3323 . . 3 (A = {B} → A {B})
2018, 19jaoi 368 . 2 ((A = A = {B}) → A {B})
2115, 20impbii 180 1 (A {B} ↔ (A = A = {B}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ∅c0 3550  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741 This theorem is referenced by:  eqsn  3867  snsssn  3873  pwsn  3881  unsneqsn  3887  foconst  5280
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