Theorem prneli 3658

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 3657	. 2 ⊢ ¬ 𝐴 ∈ {𝐵, 𝐶}
4	3	nelir 2474	1 ⊢ 𝐴 ∉ {𝐵, 𝐶}

Syntax hints:   ≠ wne 2376   ∉ wnel 2471  {cpr 3634

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640

This theorem is referenced by: (None)