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Theorem nelpri 3754
 Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)
Hypotheses
Ref Expression
nelpri.1 AB
nelpri.2 AC
Assertion
Ref Expression
nelpri ¬ A {B, C}

Proof of Theorem nelpri
StepHypRef Expression
1 nelpri.1 . 2 AB
2 nelpri.2 . 2 AC
3 neanior 2601 . . 3 ((AB AC) ↔ ¬ (A = B A = C))
4 elpri 3753 . . . 4 (A {B, C} → (A = B A = C))
54con3i 127 . . 3 (¬ (A = B A = C) → ¬ A {B, C})
63, 5sylbi 187 . 2 ((AB AC) → ¬ A {B, C})
71, 2, 6mp2an 653 1 ¬ A {B, C}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-un 3214  df-sn 3741  df-pr 3742 This theorem is referenced by: (None)
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