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	⊢ A ≠ B
nelpri.2	⊢ A ≠ C

Assertion

Ref	Expression
nelpri	⊢ ¬ A ∈ {B, C}

Proof of Theorem nelpri

Step	Hyp	Ref	Expression
1		nelpri.1	. 2 ⊢ A ≠ B
2		nelpri.2	. 2 ⊢ A ≠ C
3		neanior 2601	. . 3 ⊢ ((A ≠ B ∧ A ≠ C) ↔ ¬ (A = B ∨ A = C))
4		elpri 3753	. . . 4 ⊢ (A ∈ {B, C} → (A = B ∨ A = C))
5	4	con3i 127	. . 3 ⊢ (¬ (A = B ∨ A = C) → ¬ A ∈ {B, C})
6	3, 5	sylbi 187	. 2 ⊢ ((A ≠ B ∧ A ≠ C) → ¬ A ∈ {B, C})
7	1, 2, 6	mp2an 653	1 ⊢ ¬ A ∈ {B, C}

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)