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Theorem snsssn 3873
 Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1 A V
Assertion
Ref Expression
snsssn ({A} {B} → A = B)

Proof of Theorem snsssn
StepHypRef Expression
1 sssn 3864 . 2 ({A} {B} ↔ ({A} = {A} = {B}))
2 sneqr.1 . . . . . 6 A V
32snnz 3834 . . . . 5 {A} ≠
4 df-ne 2518 . . . . 5 ({A} ≠ ↔ ¬ {A} = )
53, 4mpbi 199 . . . 4 ¬ {A} =
65pm2.21i 123 . . 3 ({A} = A = B)
72sneqr 3872 . . 3 ({A} = {B} → A = B)
86, 7jaoi 368 . 2 (({A} = {A} = {B}) → A = B)
91, 8sylbi 187 1 ({A} {B} → A = B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  Vcvv 2859   ⊆ 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: (None)
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