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Theorem difprsnss 3846
 Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difprsnss ({A, B} {A}) {B}

Proof of Theorem difprsnss
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . 5 x V
21elpr 3751 . . . 4 (x {A, B} ↔ (x = A x = B))
3 elsn 3748 . . . . 5 (x {A} ↔ x = A)
43notbii 287 . . . 4 x {A} ↔ ¬ x = A)
5 biorf 394 . . . . 5 x = A → (x = B ↔ (x = A x = B)))
65biimparc 473 . . . 4 (((x = A x = B) ¬ x = A) → x = B)
72, 4, 6syl2anb 465 . . 3 ((x {A, B} ¬ x {A}) → x = B)
8 eldif 3221 . . 3 (x ({A, B} {A}) ↔ (x {A, B} ¬ x {A}))
9 elsn 3748 . . 3 (x {B} ↔ x = B)
107, 8, 93imtr4i 257 . 2 (x ({A, B} {A}) → x {B})
1110ssriv 3277 1 ({A, B} {A}) {B}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ⊆ wss 3257  {csn 3737  {cpr 3738 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-sn 3741  df-pr 3742 This theorem is referenced by: (None)
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