Theorem nelpri 3690

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 2487	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
4		elpri 3689	. . . 4 ⊢ (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))
5	4	con3i 635	. . 3 ⊢ (¬ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
6	3, 5	sylbi 121	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ¬ 𝐴 ∈ {𝐵, 𝐶})
7	1, 2, 6	mp2an 426	1 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}

Syntax hints:  ¬ wn 3   ∧ wa 104   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  {cpr 3667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673

This theorem is referenced by:  prneli  3691  pw1nel3  7412  sucpw1nel3  7414