Theorem prneli 3601

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using ∉. (Contributed by David A. Wheeler, 10-May-2015.)

Hypotheses

Ref	Expression
prneli.1	⊢ 𝐴 ≠ 𝐵
prneli.2	⊢ 𝐴 ≠ 𝐶

Assertion

Ref	Expression
prneli	⊢ 𝐴 ∉ {𝐵, 𝐶}

Proof of Theorem prneli

Step	Hyp	Ref	Expression
1		prneli.1	. . 3 ⊢ 𝐴 ≠ 𝐵
2		prneli.2	. . 3 ⊢ 𝐴 ≠ 𝐶
3	1, 2	nelpri 3600	. 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}
4	3	nelir 2434	1 ⊢ 𝐴 ∉ {𝐵, 𝐶}

Syntax hints:   ≠ wne 2336   ∉ wnel 2431  {cpr 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by: (None)